Systems and methods for quantifying nonlinearities in interferometry systems

ABSTRACT

The invention features interferometry systems and methods that quantify nonlinearities, e.g., cyclic errors, in an interference signal produced by an interferometry system. The systems and methods analyze interference signals for each of multiple Doppler shifts to thereby resolve nonlinearities that may otherwise overlap spectrally with a dominant interference signal, and also, to interpolate the contributions of the nonlinearities to measurements at different Doppler shifts. The time-varying interference signal or the phase extracted from the time-varying interference signal is Fourier transformed and at least some of the nonlinearities are associated with peaks in the square modulus of the Fourier transformed signal (i.e., the power spectrum). The amplitude and phase of the Fourier transform at the frequency of each such peak are used to quantify the associated nonlinearity. The quantified nonlinearities are used to correct optical path length measurements by the system. Changes in the magnitude of one or more of the quantified nonlinearities can also be used to identify degradation of a component of the interferometry system.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from provisional application 60/166,639filed Nov. 19, 1999, the contents of which are incorporated herein byreference.

BACKGROUND OF THE INVENTION

This invention relates to interferometers, e.g., displacement measuringand dispersion interferometers that measure displacements of ameasurement object such as a mask stage or a wafer stage in alithography scanner or stepper system, and also interferometers thatmonitor wavelength and determine intrinsic properties of gases.

Displacement measuring interferometers monitor changes in the positionof a measurement object relative to a reference object based on anoptical interference signal. The interferometer generates the opticalinterference signal by overlapping and interfering a measurement beamreflected from the measurement object with a reference beam reflectedfrom the reference object.

In many applications, the measurement and reference beams haveorthogonal polarizations and different frequencies. The differentfrequencies can be produced, for example, by laser Zeeman splitting, byacousto-optical modulation, or internal to the laser using birefringentelements or the like. The orthogonal polarizations allow a polarizingbeam splitter to direct the measurement and reference beams to themeasurement and reference objects, respectively, and combine thereflected measurement and reference beams to form overlapping exitmeasurement and reference beams. The overlapping exit beams form anoutput beam that subsequently passes through a polarizer. The polarizermixes polarizations of the exit measurement and reference beams to forma mixed beam. Components of the exit measurement and reference beams inthe mixed beam interfere with one another so that the intensity of themixed beam varies with the relative phase of the exit measurement andreference beams. A detector measures the time-dependent intensity of themixed beam and generates an electrical interference signal proportionalto that intensity. Because the measurement and reference beams havedifferent frequencies, the electrical interference signal includes a“heterodyne” signal having a beat frequency equal to the differencebetween the frequencies of the exit measurement and reference beams. Ifthe lengths of the measurement and reference paths are changing relativeto one another, e.g., by translating a stage that includes themeasurement object, the measured beat frequency includes a Doppler shiftequal to 2vnp/λ, where v is the relative speed of the measurement andreference objects, λ is the wavelength of the measurement and referencebeams, n is the refractive index of the medium through which the lightbeams travel, e.g., air or vacuum, and p is the number of passes to thereference and measurement objects. Changes in the relative position ofthe measurement object correspond to changes in the phase of themeasured interference signal, with a 2π phase change substantially equalto a distance change L of λ/(np), where L is a round-trip distancechange, e.g., the change in distance to and from a stage that includesthe measurement object.

Unfortunately, this equality is not always exact. Many interferometersinclude nonlinearities such as what are known as “cyclic errors.” Thecyclic errors can be expressed as contributions to the phase and/or theintensity of the measured interference signal and have a sinusoidaldependence on the change in optical path length pnL. In particular, thefirst order cyclic error in phase has a sinusoidal dependence on(2πpnL)/λ and the second order cyclic error in phase has a sinusoidaldependence on 2(2πpnL)/λ. Higher order cyclic errors can also bepresent.

Cyclic errors can be produced by “beam mixing,” in which a portion of aninput beam that nominally forms the reference beam propagates along themeasurement path and/or a portion of an input beam that nominally formsthe measurement beam propagates along the reference path. Such beammixing can be caused by ellipticity in the polarizations of the inputbeams and imperfections in the interferometer components, e.g.,imperfections in a polarizing beam splitter used to direct orthogonallypolarized input beams along respective reference and measurement paths.Because of beam mixing and the resulting cyclic errors, there is not astrictly linear relation between changes in the phase of the measuredinterference signal and the relative optical path length pnL between thereference and measurement paths. If not compensated, cyclic errorscaused by beam mixing can limit the accuracy of distance changesmeasured by an interferometer. Cyclic errors can also be produced byimperfections in transmissive surfaces that produce undesired multiplereflections within the interferometer and imperfections in componentssuch as retroreflectors and/or phase retardation plates that produceundesired ellipticities in beams in the interferometer. For a generalreference on the theoretical cause of cyclic error, see, for example, C.W. Wu and R. D. Deslattes, “Analytical modelling of the periodicnonlinearity in heterodyne interferometry,” Applied Optics, 37,6696-6700, 1998.

In dispersion measuring applications, optical path length measurementsare made at multiple wavelengths, e.g., 532 nm and 1064 nm, and are usedto measure dispersion of a gas in the measurement path of the distancemeasuring interferometer. The dispersion measurement can be used toconvert the optical path length measured by a distance measuringinterferometer into a physical length. Such a conversion can beimportant since changes in the measured optical path length can becaused by gas turbulence and/or by a change in the average density ofthe gas in the measurement arm even though the physical distance to themeasurement object is unchanged. In addition to the extrinsic dispersionmeasurement, the conversion of the optical path length to a physicallength requires knowledge of an intrinsic value of the gas. The factor Γis a suitable intrinsic value and is the reciprocal dispersive power ofthe gas for the wavelengths used in the dispersion interferometry. Thefactor Γ can be measured separately or based on literature values.Cyclic errors in the interferometer also contribute to dispersionmeasurements and measurements of the factor Γ. In addition, cyclicerrors can degrade interferometric measurements used to measure and/ormonitor the wavelength of a beam.

SUMMARY OF THE INVENTION

The invention features interferometry systems and methods that quantifynonlinearities, e.g., cyclic errors, in an interference signal. Thenonlinearities are caused by properties of the interferometry systemsuch as beam mixing, multiple reflections, and nonlinear signalprocessing electronics. The nonlinearities produce additional terms inthe interference signal that cause the phase of the interference signalto deviate from a linear relationship with the optical path lengthdifference. The systems and methods allow the accuracy of displacement,wavelength, and dispersion measurements to be improved by correcting themeasurements for the contribution of the nonlinearities. Moreover,sources of nonlinearity not previously recognized have been identifiedand formalized.

The systems and methods analyze multiple measurements of an interferencesignal corresponding to different optical path length differences toquantify the nonlinearities. In particular embodiments, the time-varyinginterference signal or the phase extracted from the time-varyinginterference signal is Fourier transformed and at least some of thenonlinearities are associated with peaks in the square modulus of theFourier transformed signal (i.e., the power spectrum). The amplitude andphase of the Fourier transform at the frequency of each such peak areused to quantify the associated nonlinearity. The frequency of each peakand whether it can be resolved typically depends on the rate of changeof the optical path length difference, i.e., on the Doppler shift. Thus,the systems and methods often analyze multiple time-varying interferencesignals for each of multiple Doppler shifts to thereby resolvenonlinearities that may otherwise remain hidden, and also, tointerpolate the contributions of the nonlinearities to measurements atdifferent Doppler shifts. For example, the contribution of thenonlinearities can be interpolated for measurements when the measurementobject is stationary or changing direction, i.e., when the Doppler shiftis zero or is passing through zero.

In general, in one aspect, the invention features an interferometrysystem. The interferometry system includes: an interferometer whichduring operation directs two beams along separate paths and thencombines the beams to produce an overlapping pair of exit beams, theseparate paths defining an optical path length difference; a detectorwhich responds to optical interference between the overlapping pair ofexit beams and produces an interference signal s(t) indicative of theoptical path length difference; and an analyzer coupled to the detector.The signal s(t) includes a dominant term having a frequency equal to thesum of the frequency splitting ω between the two beams, if any, and aDoppler shift {dot over (φ)} defined by the rate of change of theoptical path length difference. Properties of the interferometry systemcause the signal s(t) to further include additional terms each having afrequency not equal to the sum of the frequency splitting ω and theDoppler shift {dot over (φ)}. During operation the analyzer: i)quantifies at least one of the additional terms based on values of s(t)for which the value of the Doppler shift causes the dominant term andthe at least one additional term to be separated spectrally; and ii)uses the quantified at least one additional term to estimate a change inthe optical path length difference corresponding to another value ofs(t) for which the value of the Doppler shift does not causes thedominant term and the at least one additional term to overlapspectrally.

The interferometry system can include any of the following features.

The detector can include a photodetector, an amplifier, and ananalog-to-digital converter. The frequency splitting between the twobeams can be nonzero. The at least one of the additional terms caninclude a plurality of the additional terms.

To quantify the at least one additional term, the analyzer can calculatethe Doppler shift {dot over (φ)} for the values of s(t) based on theexpression s(t)∝cos(ωt+φ+ζ_(1,0,1,0))+NL, where NL is an initialquantification of the additional terms, and where φ=Lkn, L is thephysical path length difference, k is a wavenumber, n is a refractiveindex, ω is the angular frequency splitting between the two beams, t istime, and ζ_(1,0,1,0) is a phase-offset. The initial quantification canbe NL=0.

The analyzer can quantify the at least one additional term by estimatingcorresponding coefficients of a representation of s(t) that accounts forthe additional terms. For example, the representation of s(t) can beexpressed as: $\begin{matrix}{{s(t)} = \quad {{a_{1,0,1,0}{\cos \left( {{\omega \quad t} + \phi + \zeta_{1,0,1,0}} \right)}} +}} \\{\quad {{\sum\limits_{u,{u^{\prime}p},p^{+}}{a_{u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega \quad t} + {\omega_{u^{\prime}}^{\prime}t} + {p\quad \phi} - {p^{+}\phi^{+}} + \zeta_{u,u^{\prime},p,p^{+}}} \right)}}} +}} \\{\quad {{{\sum\limits_{q}{\left( a_{1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{\quad_{1,0,1,0,q,q}}{\cos \left\lbrack {{q\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{\quad_{1,0,1,0,q,{q - 2}}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}} + \ldots}\quad,}}\end{matrix}$

u=0 or 1; u′=0,1, . . . ; ω′₀=0;

p,p⁺=0,1, . . . , w_(2,1)/w_(2,2),

p⁺≠0 if p=1 and u=1,

w_(2,1),w_(2,2)=1,2, . . . , w_(2,1)≠w_(2,2),

q=2,3 . . . ,

q_(R)=0 for q even, 1 for q odd

where

φ=Lkn,

φ⁺ =Lk ⁺ n,

k=2π/λ,

k ⁺=2π[(1/λ)+(ω/2πc)],

wherein ω is the angular frequency splitting between the two beams,ω′_(u′) are frequencies not equal to ω caused by at least one of thedetector, the analyzer, and a source for the two beams, L is thephysical path length difference, λ is the wavelength of the beams in thefirst set, n is a refractive index, c is the speed of light in vacuum,and t is time. The dominant term corresponds toa_(1,0,1,0)cos(ωt+φ+ζ_(1,0,1,0)) and the additional terms correspond tothe remaining terms. The amplitudes a_(v) and B_(v) and phases ζ_(v)define the coefficients for the representation of s(t), the subscript vdenoting a general index.

To quantify the at least one additional term, the analyzer can calculatea frequency spectrum corresponding to a set of the values of s(t), andthen estimate the coefficients for the at least one additional termbased on the amplitude and phase of the frequency spectrum at an angularfrequency {tilde over (ω)} equal to the derivative with respect to timeof the argument of one of the sinusoids in the representation of s(t)not corresponding to the dominant term, or an alias {tilde over (ω)}_(A)of {tilde over (ω)}. For example, the frequency spectrum can be theFourier transform of the set of values of s(t). Alternatively, thefrequency spectrum can be the Fourier transform of α(t), where s(t) isexpressed as s(t)=A(t)cos(α(t)), and α(t) is the phase of s(t). If theanalyzer estimates the coefficients for the at least one additional termbased on the amplitude and phase of the frequency spectrum at the alias{tilde over (ω)}_(A) of {tilde over (ω)}, the alias frequency can beexpressed as {tilde over (ω)}_(A)=(−1)^(r){tilde over(ω)}−[(−1)^(r)(r+(½))−(½)]ω_(Ny) for a positive integer of r thatsatisfies rω_(Ny)<{tilde over (ω)}<(r+1)ω_(Ny), where the detector has asampling rate that defines the Nyquist frequency ω_(Ny). To estimate thecoefficients, for example, {tilde over (ω)} can be one of ω+ω′_(u′) foru′≠0; {tilde over (ω)} can be one of q(ω+{dot over (φ)}); or {tilde over(ω)} can be one of uω+p{dot over (φ)}+p⁺{dot over (φ)}, for p≠1 and, p≠0when u=0.

To estimate the coefficients for the at least one additional term, theanalyzer can also normalize the amplitude and phase of the frequencyspectrum at the angular frequency {tilde over (ω)} to account for atleast one non-zero, derivative of {dot over (φ)}.

The analyzer can quantify the at least one additional term based on afirst set of values of s(t) for which the Doppler shift is sufficientlylarge to spectrally separate the additional frequency from the dominantfrequency, and then further quantify the at least one additional termbased on at least a second set of values of s(t) for which the Dopplershift is different from that of the first set of values and sufficientlylarge to spectrally separate the additional frequency from the dominantfrequency. The analyzer can then quantify the at least one additionalterm as a function of the Doppler shift by interpolating values of thequantification for each set of values of s(t).

The analyzer can determine the dependence of each of the estimatedcoefficients on the Doppler shift based on multiple sets of values ofs(t), each set corresponding to a different Doppler shift.

The at least one additional term can be a plurality of the additionalterms, to quantify the plurality of the additional terms, the analyzercan estimate the coefficients for each of the plurality of theadditional terms based on the amplitude and phase of the frequencyspectrum at a corresponding plurality of angular frequencies {tilde over(ω)}_(v) or their aliases. Each {tilde over (ω)}_(v) equals thederivative with respect to time of the argument of one of the sinusoidsin the representation of s(t) not corresponding to the dominant term. Insuch embodiments, the analyzer can estimate coefficients correspondingto at least some of B_(1,0,1,0,q,q−2j) and ζ_(1,0,1,0,q,q−2j), where qis odd and j is a nonnegative integer less than q/2−1, to determineB_(1,0,1,0,q,1) and ζ_(1,0,1,0,q,1) (e.g., zero-frequency-shift errors).

The analyzer can estimate the change in the optical path lengthdifference corresponding to the other value of s(t) by determining avalue for φ=Lkn that is self-consistent withs(t)∝cos(ωt+φ+ζ_(1,0,1,0))+NL(φ,{dot over (φ)}). NL expresses thequantified at least one additional term, wherein L is the physical pathlength difference, k is a wavenumber, n is a refractive index, ω is theangular frequency difference between the two beams, t is time, andζ_(1,0,1,0) is a phase-offset. For example, the analyzer can determinethe value for φ by iteratively improving an estimate for the value forφ.

The analyzer can use the estimated change in optical path length todetermine a change in physical path length, to determine a change indispersion, to determine an intrinsic value a gas, or to monitor thewavelength of the beams.

In general, in another aspect, the invention features an interferometrysystem including: an interferometer which during operation directs twobeams along separate paths and then combines the beams to produce anoverlapping pair of exit beams, the separate paths defining an opticalpath length difference; a detector which responds to opticalinterference between the overlapping pair of exit beams and produces asignal s(t) indicative of the interference; and an analyzer coupled tothe detector. The signal s(t) is a function of the optical path lengthdifference. Properties of the interferometry system cause the signals(t) to deviate from the expression s(t)=a cos(ωt+φ+ζ), where φ=Lkn, Lis the physical path length difference, k is a wavenumber, n is arefractive index, ω is the angular frequency difference, if any, betweenthe two beams, t is time, a is an amplitude that is constant withrespect to φ, and ζ is a phase offset that is constant with respect to φand {dot over (φ)}. During operation, the analyzer: i) Fouriertransforms at least one set of values of s(t) for which the rate ofchange of the optical path length difference is not zero ({dot over(φ)}≠0), the Fourier transform defining a power spectrum equal to thesquare modulus of the Fourier transform; ii) quantifies at least some ofthe deviations based on the amplitude and phase of the Fourier transformat frequencies that differ from ω+{dot over (φ)} and correspond to peaksin the power spectrum; and iii) uses the quantified deviations toestimate a change in the optical path length difference corresponding toa particular value of s(t).

In general, in another aspect, the invention features an interferometrysystem including: an interferometer which during operation directs twobeams along separate paths and then combines the beams to produce anoverlapping pair of exit beams, the separate paths defining an opticalpath length difference; a detector which responds to opticalinterference between the overlapping pair of exit beams and produces asignal s(t) indicative of the interference; and an analyzer coupled tothe detector. The signal s(t) is a function of the optical path lengthdifference. Properties of the interferometry system cause the signals(t) to deviate from the expression s(t)=a cos(ωt+φ+ζ), where φ=Lkn, Lis the physical path length difference, k is a wavenumber, n is arefractive index, ω is the angular frequency difference, if any, betweenthe two beams, t is time, a is an amplitude that is constant withrespect to φ, and ζ is a phase offset that is constant with respect to φand {dot over (φ)}, where s(t) can be expressed as s(t)=A(t)cos(α(t)),and α(t) is the phase of s(t). During operation, the analyzer: i)extracts the phase α(t) for s(t); ii) Fourier transforms at least oneset of values of α(t) for which the rate of change of the optical pathlength difference is not zero ({dot over (φ)}≠0), the Fourier transformdefining a power spectrum equal to the square modulus of the Fouriertransform; iii) quantifies at least some of the deviations based on theamplitude and phase of the Fourier transform at frequencies that differfrom ω+{dot over (φ)} and correspond to peaks in the power spectrum; andiv) uses the quantified deviations to estimate a change in the opticalpath length difference corresponding to a particular value of s(t).

In general, in another aspect, the invention features an interferometrysystem including: an interferometer which during operation directs twobeams along separate paths and then combines the beams to produce anoverlapping pair of exit beams, the separate paths defining an opticalpath length difference; a detector which responds to opticalinterference between the overlapping pair of exit beams and produces aninterference signal s(t) indicative of the optical path lengthdifference; an analyzer coupled to the detector; and an alert mechanismcoupled to the analyzer. The signal s(t) includes a dominant term havinga frequency equal to the sum of the frequency splitting ω between thetwo beams, if any, and a Doppler shift {dot over (φ)} defined by therate of change of the optical path length difference. Properties of theinterferometry system cause the signal s(t) to further includeadditional terms each having a frequency not equal to the sum of thefrequency splitting ω and the Doppler shift {dot over (φ)}. Duringoperation, the analyzer monitors the frequencies of the signal s(t), andproduces a signal indicative of system degradation when the amplitude ofa frequency corresponding to one of the additional terms exceeds athreshold value. The alert mechanism is responsive to the systemdegradation signal.

The interferometry system can include any of the following features.

To alert an operator, the alert mechanism can include at least one of avisual display, a sound system, a warning light, and a printer.

To monitor the frequencies in s(t), the analyzer can Fourier transformat least one set of values for s(t). Alternatively, the analyzer canextract the phase α(t) of s(t) for a set of values of s(t), where s(t)is expressed as s(t)=A(t)cos(α(t)), and then Fourier transform theextracted phases to monitor the frequencies in s(t).

The analyzer can monitor the frequencies in s(t) based on values of s(t)for which the value of the Doppler shift causes the dominant term and atleast one of the additional terms to be separated spectrally.

The signal s(t) can be expressed by the summation shown above withreference to an earlier aspect of the invention. To determine whether toproduce the signal indicative of system degradation, the analyzer cancompare the threshold value to the amplitude of one of frequenciesω+ω′_(u′), for u′≠0, frequencies q(ω+{dot over (φ)}), or frequenciesuω+p{dot over (φ)}+p⁺{dot over (φ)}, for p≠1, and for p≠0 when u=0.

In general, in another aspect, the invention features an interferometrysystem including: a source which during operation provides a first setof two beams having a frequency splitting ω and a second set of twobeams having a frequency splitting ω_(T) not equal to ω; aninterferometer which during operation directs the first beam of thefirst set and the first beam of the second set along a measurement pathand the second beam of the first set and the second beam of the secondset along a reference path, and then combines the two sets of beams toform an output beam, the measurement and reference paths defining anoptical path length difference; a detector which responds to opticalinterference between the beams in the output beam and produces a signalS(t) indicative of the interference, the interference being a functionof the optical path length difference; and an analyzer coupled to thedetector. In the absence of the second set of beams, the signal S(t)equals s(t) which includes a dominant term at a frequency equal to thesum of the frequency splitting ω and a Doppler shift {dot over (φ)}defined by the rate of change of the optical path length difference.Properties in the interferometry system cause zero-frequency-shiftcyclic errors that contribute to s(t) at the same frequency as thedominant frequency. In the presence of the second set of beams, theproperties that produce the zero-frequency-shift cyclic errorcontribution to s(t) produce a multiplet in the frequency spectrum ofS(t), wherein the multiplet has adjacent peaks that are spaced byω−ω_(T). During operation the analyzer resolves frequencies in S(t) toidentify the multiplet and quantifies at least one of thezero-frequency-shift cyclic errors based on the amplitude and phase ofat least one of the peaks in the multiplet.

The interferometry system can include any of the following features.

The analyzer can quantify multiple zero-frequency-shift cyclic errorsbased on the amplitude and phase of each of multiple peaks in themultiplet. The analyzer can be further coupled to the source, and canselectively cause the source to provide the first set of beams and notthe second set of beams to the interferometer. When the analyzerselectively causes the source to provide the first set of beams and notthe second set of beams to the interferometer, the analyzer candetermine the optical path length difference based on s(t) and at leastone of the quantified zero-frequency-shift cyclic errors. Alternatively,the analyzer can determine the optical path length difference based onS(t) and at least one of the quantified zero-frequency-shift cyclicerrors.

The analyzer can resolve frequencies in S(t) by Fourier transforming atleast one set of values for S(t). Alternatively, S(t) is expressed asS(t)=A_(S)(t)cos(α_(S)(t)), α_(S)(t) being the phase of S(t), and theanalyzer can resolve the frequencies of S(t) by extracting the phaseα_(S)(t) from S(t) and Fourier transforming at least one set of valuesof α_(S)(t).

The multiplet can include a peak at the dominant frequency. Thefrequency splittings can each be less than a Nyquist frequency ω_(Ny),where the detector samples values of S(t) at a rate that defines theNyquist frequency. The difference between the average frequency of thefirst set of beams and the average frequency of the second set of beamscan be more than the Nyquist frequency. For example, the frequencysplittings can satisfy: ω<ω_(Ny), ω_(T)<ω_(Ny), and |ω−ω_(T)|<<ω, e.g.,|ω−ω_(T)|<(ω/100).

The source can include first and second lasers, the first set of beamsderived from the first laser and the second set of beams derived fromthe second laser. Furthermore, the source can include first and secondlasers and first and second acousto-optical modulators, the first set ofbeams derived from the first laser and the first acousto-opticalmodulator and the second set of beams derived from second laser and thesecond acousto-optical modulator. Alternatively, the source includes alaser and first and second acousto-optical modulators, wherein a firstbeam derived from the laser passes through first acousto-opticalmodulator to produce the first set of beams and a second beam derivedfrom the laser passes through the second acousto-optical modulator toproduce the second set of beams. For example, the first and second beamsderived from the laser can correspond to adjacent longitudinal modes ofthe laser.

The analyzer can resolve the frequency multiplet in S(t) for each ofmultiple Doppler shifts and quantify the dependence of the quantifiedzero-frequency-shift cyclic on the Doppler shift. During operation theanalyzer can produce a signal indicative of system degradation when theamplitude of the multiplet exceeds a threshold value. The system canfurther include an alert mechanism coupled to the analyzer that isresponsive to the system degradation signal. For example, the alertmechanism can include at least one of a visual display, a sound speaker,a printer, and a warning light.

Finally, the signal s(t) can be expressed by the summation shown abovewith reference to an earlier aspect of the invention, in which case thequantified zero-frequency-shift cyclic error can correspond toB_(1,0,1,q,1) and ζ_(1,0,1,q,1) for one of q=3,5,7 . . . .

In general, in another aspect, the invention features a lithographysystem for use in fabricating integrated circuits on a wafer, the systemincluding: a stage for supporting the wafer; an illumination system forimaging spatially patterned radiation onto the wafer; a positioningsystem for adjusting the position of the stage relative to the imagedradiation; and any of the interferometry systems described above formeasuring the position of the stage.

In general, in another aspect, the invention features a lithographysystem for use in fabricating integrated circuits on a wafer, the systemincluding: a stage for supporting the wafer; and an illumination systemincluding a radiation source, a mask, a positioning system, a lensassembly, and any of the interferometry systems described above, whereinduring operation the source directs radiation through the mask toproduce spatially patterned radiation, the positioning system adjuststhe position of the mask relative to the radiation from the source, thelens assembly images the spatially patterned radiation onto the wafer,and the interferometry system measures the position of the mask relativeto the radiation from the source.

In general, in another aspect, the invention features a beam writingsystem for use in fabricating a lithography mask, the system including:a source providing a write beam to pattern a substrate; a stagesupporting the substrate; a beam directing assembly for delivering thewrite beam to the substrate; a positioning system for positioning thestage and beam directing assembly relative one another; and any of theinterferometry systems describe for measuring the position of the stagerelative to the beam directing assembly.

In further aspects, the invention features interferometry method,lithography methods, and beam writing methods based on the systemsdescribed above. General aspects of such methods are described below.

In one aspect, the invention features an interferometry method for usewith an interferometry system. The interferometry method includes:directing two beams along separate paths; combining the beams to producean overlapping pair of exit beams, the separate paths defining anoptical path length difference; measuring optical interference betweenthe overlapping pair of exit beams to produce an interference signals(t) indicative of the optical path length difference, the signal s(t)including a dominant term having a frequency equal to the sum of thefrequency splitting ω between the two beams, if any, and a Doppler shift{dot over (φ)} defined by the rate of change of the optical path lengthdifference, wherein properties of the interferometry system causes thesignal s(t) to further include additional terms each having a frequencynot equal to the sum of the frequency splitting ω and the Doppler shift{dot over (φ)}; quantifying at least one of the additional terms basedon values of s(t) for which the value of the Doppler shift causes thedominant term and the at least one additional term to be separatedspectrally; and using the quantified at least one additional term toestimate a change in the optical path length difference corresponding toanother value of s(t) for which the value of the Doppler shift does notcauses the dominant term and the at least one additional term to overlapspectrally.

In another aspect, the invention features an interferometry method foruse with an interferometry system. The interferometry method includes:directing two beams along separate paths; combining the beams to producean overlapping pair of exit beams, the separate paths defining anoptical path length difference; measuring optical interference betweenthe overlapping pair of exit beams to produce an interference signals(t) indicative of the optical path length difference, the signal s(t)including a dominant term having a frequency equal to the sum of thefrequency splitting ω between the two beams, if any, and a Doppler shift{dot over (φ)} defined by the rate of change of the optical path lengthdifference, wherein properties of the interferometry system causes thesignal s(t) to further include additional terms each having a frequencynot equal to the sum of the frequency splitting ω and the Doppler shift{dot over (φ)}; monitoring the frequencies of the signal s(t); andalerting an operator when the amplitude of a frequency corresponding toone of the additional terms exceeds a threshold value.

In another aspect, the invention features an interferometry method foruse with an interferometry system. The interferometry method includes:providing a first set of two beams having a frequency splitting ω and asecond set of two beams having a frequency splitting ω_(T) not equal toω; directing the first beam of the first set and the first beam of thesecond set along a measurement path and the second beam of the first setand the second beam of the second set along a reference path; combiningthe two sets of beams to form an output beam, the measurement andreference paths defining an optical path length difference; measuringoptical interference between the beams in the output beam to produce asignal S(t) indicative of the interference, the interference being afunction of the optical path length difference, wherein in the absenceof the second set of beams, the signal S(t) equals s(t) which includes adominant term at a frequency equal to the sum of the frequency splittingω and a Doppler shift {dot over (φ)} defined by the rate of change ofthe optical path length difference, wherein properties in theinterferometry system cause zero-frequency-shift cyclic errors thatcontribute to s(t) at the same frequency as the dominant frequency, andin the presence of the second set of beams, the properties that producethe zero-frequency-shift cyclic error contribution to s(t) produce amultiplet in the frequency spectrum of S(t), wherein the multiplet hasadjacent peaks that are spaced by ω−ω_(T); resolving frequencies in S(t)to identify the multiplet; and quantifying at least one of thezero-frequency-shift cyclic errors based on the amplitude and phase ofat least one of the peaks in the multiplet.

In a further aspect, the invention features a lithography methodincluding: supporting a wafer on a stage; imaging spatially patternedradiation onto the wafer; adjusting the position of the stage relativeto the imaged radiation; and using any of the interferometry methodsdescribed above to measure the relative position of the stage.

In another aspect, the invention features a lithography methodincluding: supporting a wafer on a stage; directing radiation from asource through a mask to produce spatially patterned radiation;positioning the mask relative to the radiation; using any of theinterferometry methods described above to measures the position of themask relative to the radiation; and imaging the spatially patternedradiation onto the wafer.

In another aspect, the invention features a beam writing methodincluding: providing a write beam to pattern a substrate; supporting thesubstrate on a stage; delivering the write beam to the substrate;positioning the stage relative to the write beam; and using any of theinterferometry methods describe above to measure the relative positionof the stage.

Embodiments of the invention can include many advantages. For example,they can identify and quantify nonlinearities that can otherwise degradethe interferometric displacement or dispersion measurement. Thequantified nonlinearities can be used to correct interferometricmeasurements and thereby significantly improve their accuracy. Moreover,by using the systems of and methods of the invention, interferometerscan be made more cheaply because expensive optical components thatreduce the likelihood of nonlinearities are not necessary, similarly,nonlinearities in the detection electronics need not be minimized. Inaddition, by using the systems and methods of the invention, adegradation in performance of one or more components of aninterferometer can be detected and corrective measures implemented, forexample, as part of a programmed maintenance, thus reducing thepotential for a significant loss in acceptable operation time as aconsequence of operating the interferometer in an unacceptable mode. Byquantifying the nonlinearities, embodiments of the invention can permitrapid correction of the interferometric measurement, such as is usuallynecessary during online applications when the measurement object israpidly scanned or stepped. The quantification of the nonlinearities andits use in correcting interferometric measurements can be applied tooptical distance measurements, dispersion measurements, wavelengthmeasurements, and measurements of intrinsic optical properties of thegas in the measurement arm of the interferometer such as the reciprocaldispersive power Γ. In addition, the interferometry systems can be usedin lithography and mask writing applications.

Other features and advantages will be apparent from the followingdetailed description and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general schematic diagram of an interferometry system thatquantifies and/or monitors nonlinearities caused by properties of thesystem.

FIG. 2a is schematic diagrams of a first embodiment of an interferometrysystem that quantifies nonlinearities. FIGS. 2b and 2 c are schematicdiagrams of various embodiments for an electronic processor in theinterferometry system of FIG. 2a. FIG. 2d is a graph that illustratesthe frequencies of various types of nonlinearities. FIGS. 2e and 2 f areschematic diagrams of different embodiments for a source to produce totwo input beams for the interferometry system of FIG. 2a to quantifyzero-frequency-shift cyclic errors.

FIG. 3a is schematic diagrams of a second embodiment of aninterferometry system that quantifies nonlinearities. FIGS. 3b and 3 care schematic diagrams of various embodiments for an electronicprocessor in the interferometry system of FIG. 3a.

FIG. 4a is schematic diagrams of a third embodiment of an interferometrysystem that quantifies nonlinearities. FIGS. 4b and 4 c are schematicdiagrams of electronic processors in the interferometry system of FIG.4a.

FIGS. 5a and 5 b are schematic diagrams of electronic processors for usewith the interferometry system of FIG. 4a in a fourth embodiment of theinvention. FIGS. 5c and 5 d are schematic diagrams of electronicprocessors for use with the interferometry system of FIG. 4a in a fifthembodiment of the invention.

FIG. 6a is schematic diagram of a lithography system that includes aninterferometry system described herein and is used to make integratedcircuits. FIGS. 6b-c are flow charts that described steps for makingintegrated circuits.

FIG. 7 is a schematic of a beam writing system that includes aninterferometry system described herein.

DETAILED DESCRIPTION

Nonlinearities such as cyclic errors can degrade the accuracy ofdisplacement and/or dispersion measurements extracted frominterferometric data. The nonlinearities can arise from imperfections inthe light source and the interferometer optics, and from nonlinearity inthe detection electronics such as the photoelectric detector, amplifier,or analog-to-digital converter. Although it might be possible tominimize the causes of such nonlinearities, one aspect of the presentinvention proposes to improve the accuracy of the interferometricmeasurement by quantifying the nonlinearities and, using the quantifiednonlinearities, correcting the interferometric signal (or informationderived from the interferometric signal) for the nonlinearities tothereby improve the accuracy of the measurement of interest, e.g.,displacement or dispersion. Another aspect of the present inventionproposes to detect a degradation of certain components of aninterferometer system by quantifying nonlinearities of theinterferometer system, and monitoring whether the components aredegrading based on a change in the magnitude of the quantifiednonlinearities. Interferometry systems that provide such features willnow be described generally, and thereafter, more specific embodimentswill be described in greater detail.

Referring to FIG. 1, an interferometry system 10 includes a source 20,an interferometer 30, a detector 40, and an analyzer 50. Source 20includes a laser for providing one or more beams 25 to interferometer30. For dispersion interferometry, beams 25 include at least two beamsat different wavelengths, e.g., 1064 nm and 532 nm. For optical pathdisplacement measurements, a single wavelength is sufficient. When usingheterodyne interferometry techniques at one or more differentwavelengths, source 20 introduces a frequency splitting betweencomponents of each beam at the one or more different wavelengths. Forexample, one or more acousto-optical modulators can be used to introducethe frequency splitting, or alternatively, the source can include aZeeman-split laser to produce the frequency splitting. Often thefrequency-split components are made to have orthogonal polarizations.The frequency-split components can be sent to interferometer 30, wherethey are separated into measurement and reference beams. Alternatively,source 20 can spatially separate the frequency-split components and sendthe spatially separated components to interferometer 30, where theybecome measurement and reference beams.

Interferometer 30 can be any type of interferometer, e.g., adifferential plane mirror interferometer, a double-pass interferometer,or a Michelson-type interferometer. The interferometer can be designedto monitor, for example, changes in optical path length, changesphysical path length, changes in refractive index, changes in wavelengthof a beam, or intrinsic gas properties along a path length. Theinterferometer directs a reference beam along a reference path (whichmay contact a reference object) and a measurement beam along ameasurement path contacting a measurement object (e.g., a lithographystage), and then combines the reference and measurement beams to form anoverlapping pair of exit beams 35. In dispersion interferometryapplications, there are overlapping pairs of exit beams for each of thedifferent wavelengths.

The interference between the overlapping pair of exit beams containsinformation about the relative difference in optical path length betweenthe reference and measurement paths. In some embodiments, the referencepath is fixed and therefore changes in the optical path lengthdifference correspond to changes in the optical path length of themeasurement path. In other embodiments, however, the optical path lengthof both the reference and measurement paths can be changing. Forexample, the reference path can contact a reference object (e.g., acolumn reference), that may move relative to the interferometer. In thislatter case, the changes in the optical path length differencecorrespond to changes in the position of the measurement object relativeto the reference object.

When the reference and measurement beams have orthogonal polarizations,the intensity of at least one intermediate polarization of theoverlapping pair of exit beams is selected to produce the opticalinterference. For example, a polarizer can be positioned withininterferometer 30 to mix the polarizations of the overlapping pair ofexit beams, which is then sent to detector 40. Alternatively, thepolarizer can be positioned within detector 40. The detector 40 measuresthe intensity of the selected polarization of the overlapping pair ofexit beams to produce the interference signal. Portions of the beams canbe combined with one another before being directed along the referenceand measurement paths to provide a reference pair of overlapping exitbeams, which is used to provide a reference interference signal.

Detector 40 includes a photodetector, which measures the intensity ofthe selected polarization of the overlapping pair of exit beams, andelectronics such as a preamplifier and an analog-to-digital converter,which amplify the output from the photodetector and produce a digitalsignal s(t) corresponding to the optical interference. In dispersioninterferometry applications, digital signals s(t) are produced for eachof the overlapping pair of exit beams (which correspond to differentwavelengths) by using multiple photodetection channels within detector40.

The signal s(t), absent any nonlinearities and ignoring a constantoffset intensity, can be expressed as s(t)=a cos(ωt+φ+ζ), where φ=Lkn, Lis the physical path length difference between the reference andmeasurement paths, k is the wavenumber of the measurement beam, n is therefractive index within the interferometer, ω is the angular splitfrequency difference between the measurement and reference beams beforeintroduction of any Doppler shift, t is time, a is an amplitude that isconstant with respect to φ, and ζ is a phase offset that is constantwith respect to φ and {dot over (φ)}, where {dot over (φ)} is the firstderivative of φ with respect to time. In homodyne applications, thesplit frequency difference between beam components in the expression fors(t) is zero, i.e. ω=0, and to accurately separate background signalfrom the optical inteference, detector 40 includes multiplephotodetection channels to measure interference for multiple phaseoffsets, the phase offsets being introduced within detector 40.

The signal s(t) is sent to analyzer 50, which extracts phase φ=Lkn froms(t) using a reference phase provided by the source of the heterodynefrequency split difference or the reference interference signal, theanalyzer can determine changes in the optical length difference betweenthe measurement and reference paths. Furthermore, using the signalscorresponding to additional wavelengths, the analyzer can makedispersion measurements, determine physical path length differencemeasurements, and/or measure intrinsic properties of the gas in themeasurement path.

Analyzer 50 includes a computer or digital processor for performing thephase extraction and other analysis steps described below relating toquantification of nonlinearities. For example, the numerical andsymbolic steps described herein can be converted into a digital programexecuted, e.g., on a digital signal processor (DSP) according to methodswell known in the art. The digital program can be stored on a computerreadable medium such as a hard disk and can be executable by thecomputer processors in the analyzer. Alternatively, the appropriateanalysis steps can be converted into a digital program that is hardwiredinto dedicated electronic circuits within the analyzer that executes thesteps. Methods for generating such dedicated electronic circuits basedon a given numerical or symbolic analysis procedure are also well knownin the art.

Beam mixing and intensity fluctuations in beam 25, imperfections ininterferometer 30, and nonlinearity in detector 40 and the electronicstherein, can all produce nonlinearities in the signal s(t). Thenonlinearities cause the signal s(t) to deviate from the expressions(t)=a cos(ωt+φ+ζ), e.g., the cyclic errors described above canintroduce additional terms to the expression such as a_(p1)cos(ωt+pφ+ζ_(p1)) where p=2,3, . . . . Moreover, p can take onfractional values when there are multiple passes within interferometer30. Intensity fluctuations in beam 25 can introduce additional terms tothe expression such as a_(u′p1) cos(ωt+ω′_(u′)t+pφ+ζ_(u′p1)) whereu′=0,1, . . . . The angular frequencies ω_(u′) can arise, for example,from the switching frequencies of power supplies in source 20. Inheterodyne applications, beam mixing can also produce additional termssuch as a_(p0) cos(pφ+ζ₁₀), and there can also be terms that address thesmall wavevector difference between the measurement and reference beamsassociated with the heterodyne frequency splitting. Moreover,nonlinearity in the output and frequency response of detector 40 and theelectronics therein introduces additional terms and mixes the dominantterm a cos(ωt+φ+ζ) with the terms described above. For example, anexpression for s(t) that accounts for some of the nonlinearities cantake the form:${s(t)} = {\sum\limits_{q}{B_{q}\left\{ {\sum\limits_{u,p}{a_{up}{\cos \left( {{u\quad \omega \quad t} + {p\quad \phi} + \zeta_{up}} \right)}}} \right\}^{q}}}$

where p=1,2,3 . . . and fractional values, u=0 or 1, and q=1,2,3 . . . ,and where the “q” index is associated with nonlinearity in detector 40.The above equation for s(t), however, assumes that the nonlinearity inthe detector is frequency independent. If it is not, each of the termsthat result from the expansion in the above equation can include aphase-shift and amplitude that depend on the frequency of that term. Theexpression can be further complicated by the finite sampling rate of theanalog-to-digital converter in detector 40, which can cause aliasing inthe digital representation of s(t).

If not accounted for, the contributions of the nonlinearities to s(t)can degrade the accuracy of the optical path difference information tobe extracted from the interference signal. Often, the degree to whichthe accuracy is degraded depends on {dot over (φ)}, or the relativespeed of the measurement and reference objects, e.g., the Doppler shift.For example, at large Doppler shifts, determining a change in φ from thephase of the Fourier transform of s(t) at ω+{dot over (φ)} minimizes thecontributions from those nonlinearities that have peaks separated infrequency space from ω+{dot over (φ)}, e.g., at 2{dot over (φ)}, ω+2{dotover (φ)}, ω+3{dot over (φ)}, 2ω+2{dot over (φ)}. However, at smallerDoppler shifts, the contributions from many nonlinearities will overlapwith the dominant peak at ω+{dot over (φ)} in the power spectrum ofs(t), the power spectrum being the square modulus of the Fouriertransform of s(t). The overlap is particularly large when {dot over(ω)}=0, e.g., when the relative position of the measurement andreference object is stationary, or when the relative speed of themeasurement and reference objects changes sign. Moreover, even when theDoppler shift separates the frequency of the nonlinearity from that ofthe dominant peak in the power spectrum of s(t), aliases of thefrequency of the nonlinearities may overlap with the dominant peak atω+{dot over (φ)}. Furthermore, the magnitude and phase of thenonlinearity, e.g., B_(q)a_(up) and ζ_(up), may also vary with {dot over(φ)} because, for example, of a frequency-dependent response of thedetection electronics.

Furthermore, some of the nonlinearities have frequencies that overlapexactly with the dominant peak, regardless of the value of {dot over(φ)}. Such nonlinearities can be called zero-frequency-shift cyclicerrors. For example, in the expansion of s(t) in the equation above forq=3, the B₃a₁₁ ³ cos³(ωt+φ+ζ₁₁) produces a term at the dominantfrequency. Similarly, for example, difference mixing in the q=2expansion between a u=1,p=2 term and a u=0,p=1 term produces a term atthe dominant frequency.

Analyzer 50 quantifies the nonlinearities based on values of s(t) formultiple optical path length differences. In some embodiments, thenonlinearities are expressed as additional sinusoidal terms in anexpression for s(t) as shown, for example, in the above equation. Inother embodiments, the nonlinearities are expressed in an additionalphase term ψ in s(t), where s(t)=A(t)cos(ωt+φ+ψ+ζ) and the phase term ψcan be expressed as a series of sinusoids having arguments similar tothose shown in the sinusoids of the above equation for s(t). In eithercase, each nonlinearity is quantified by estimating the amplitude andphase of its corresponding sinusoid, the amplitude and phase definingcoefficients for the nonlinearity. Alternatively, coefficients can bedefined by the amplitudes of sine and cosine terms that both have theargument corresponding to the nonlinearity.

During operation, analyzer 50 determines φ (relative to the offset phaseζ₁₁) from the interference signal s(t) and the quantified nonlinearitiesor an initial guess for the quantified nonlinearities by using aniterative process, e.g., by first determining φ assuming nononlinearities, and then determining iteratively improved values for φfrom the interference signal by including the contributions ofnonlinearities that correspond to the previously determined value of φ.Analyzer can also determine values for {dot over (φ)} based on theinterference signal s(t) and the quantified nonlinearities or an initialguess for the quantified nonlinearities.

In embodiments where the nonlinearities are expressed as a series ofsinusoids in an expression s(t), analyzer 50 estimates the coefficientsof the nonlinearities by Fourier transforming values of s(t)corresponding to a substantially constant value of {dot over (φ)} thatcauses the frequency peaks of one or more of the nonlinearities to beseparated spectrally from the dominant frequency ω+{dot over (φ)},thereby allowing the nonlinearities to be quantified. The analyzeridentifies the dominant peak at ω+{dot over (φ)}, associates each of theremaining peaks with a nonlinearity, and determines the coefficients foreach nonlinearity from the complex amplitude of its corresponding peakand, where necessary, a normalization factor that accounts for theeffect of higher order derivatives of {dot over (φ)} on the Fouriertransform. Analyzer 50 can repeat these steps for additional sets ofvalues of s(t) corresponding to the same substantially constant value of{dot over (φ)}, and average (or “filter”) the determined coefficientsfrom all of the sets to improve the quantification of thenonlinearities.

Analyzer 50 then repeats the steps in the preceding paragraph for valuesof s(t) corresponding to a different substantially constant value of{dot over (φ)}. The Fourier transform of such values can produce peaksfor nonlinearities that were not resolved in the Fourier transform ofthe values of s(t) corresponding to the first substantially constantvalue of {dot over (φ)}, and permits the analyzer to determine thecoefficients for the previously unresolved nonlinearities. The steps canbe further repeated for values of s(t) corresponding to each ofadditional, substantially constant values of {dot over (φ)}.Furthermore, analyzer 50 can interpolate the values of the coefficientsdetermined for values of s(t) corresponding to each of the different,substantially constant values of {dot over (φ)}, to determine thedependence of each nonlinearity on {dot over (φ)}, if any.

In other embodiments in which the nonlinearities are expressed as aseries of sinusoids in the phase ψ of s(t)=A(t)cos(ωt+φ+ψ+ζ), analyzer50 estimates the coefficients of the nonlinearities by Fouriertransforming the phase α (where α=ωt+φ+ψ+ζ) of values of s(t)corresponding to each of multiple, substantially constant values of {dotover (φ)}. Otherwise the analysis is similar to that described above. Infurther embodiments, any combination of ωt, {tilde over (φ)}, and {tildeover (ψ)}, can be subtracted from α before taking the Fouriertransforms, where {tilde over (φ)} and {tilde over (ψ)} are approximateguesses for φ and ψ, respectively.

In dispersion applications or applications in which the intrinsicrefractive properties of a gas are being measured, detector 40 sendssignals s_(λ)(t) for each of multiple wavelengths λ to analyzer 50.Quantification of the nonlinearities by analyzer 50 can be based on oneor more of the signals s_(λ)(t) in a manner similar to that summarizedabove. Furthermore, because of the improved accuracy provided by thequantification of nonlinearities, interferometry system 10 can beadvantageously used in microlithography and beam-writing systems.

The magnitudes of the nonlinearities can change over time as componentsof the interferometry system degrade. For example, optical andelectronic components can degrade over time because of, e.g., overuse,faulty design, or environmental factors such as humidity, dust, andtemperature. Furthermore, environmental disturbances may degrade theoptical alignment of the system.

Referring again to FIG. 1, to identify such degradation, analyzer 50monitors the quantified nonlinearities over time to determine whetherthere is any sudden or gradual increase in the magnitude of one or moreof the quantified nonlinearities. For example, analyzer can monitor thefrequency spectrum of either s(t) or the phase α of s(t), and determinewhen peaks at frequencies other than the dominant frequency ω+{dot over(φ)} exceed an acceptable threshold level. If so, analyzer 50 sends asignal 55 indicative of system degradation to an alert mechanism 60. Thealert mechanism responds to signal 55 indicative of system degradationby alerting a user that one or more components of the interferometrysystem may have degraded beyond an acceptable level. For example, alertmechanism 60 can include one or more of a video monitor that displays anerror message in response to signal 55, a sound system or siren thatproduces an audio warning signal in response to signal 55, a printerthat prints an error message in response to signal 55, and a light thatflashes or changes color in response to signal 55. Alert mechanism 60can also be coupled to a related system, e.g., a lithography or a beamwriting system, and can cause the related system to shut down inresponse to signal 55.

The threshold level in analyzer 50 can be preset by an operator todefine the acceptable level for a particular application. Moreover, anoperator may preset multiple threshold levels, each corresponding to aparticular nonlinearity, e.g., the corresponding frequency of thenonlinearity in the frequency spectrum of s(t) or the phase α of s(t).Furthermore, even for a particular nonlinearity, the analyzer 50 cancompare the magnitude of the nonlinearity to multiple threshold levelsand cause signal 55 to indicate a degree of degradation corresponding towhich threshold levels had been exceeded. Depending on the embodiment,analyzer 50 can quantify the nonlinearities and correct the measurementof the optical path length difference using the quantifiednonlinearities, monitor system degradation based on the magnitude ofnonlinearities in the frequency spectrum of s(t) (or the phase α ofs(t)), or both.

Detailed descriptions of specific embodiments follow below. While theydiffer in some details, the disclosed embodiments otherwise share manycommon elements and naturally fall into several different categoriesdepending on the type of end use application and on the type ofprocedure used for measuring and correcting for effects ofnonlinearities due to the cyclic errors.

A first category of embodiments of the several different categoriescomprise distance measuring interferometers operating with onewavelength and effects of cyclic errors are determined and compensated.The effects of cyclic errors are determined from analyses of Fouriertransforms of electrical interference signals wherein the electricalinterference signals are generated by detection of a polarization mixedreference and measurement beams from the interferometers.

A second category of embodiments of the several different categoriescomprise distance measuring interferometers operating with onewavelength and the effects of cyclic errors are compensated usingmeasured effects of cyclic error determined in part from analyses ofFourier transforms of phases of electrical interference signals. Theelectrical interference signals are generated by detection ofpolarization mixed reference and measurement beams from theinterferometers.

Embodiments of a third category of the several different categoriescomprise an apparatus and method for compensating for effects of cyclicerrors on dispersion or dispersion and distance measuring relatedsignals. The effects of the cyclic errors are determined from analysesof Fourier transforms of electrical interference signals wherein theelectrical interference signals are generated by detection ofpolarization mixed reference and measurement beams from dispersionmeasuring and distance measuring interferometers. The effects of a gasin the measuring path of a distance measuring interferometer arecorrected by a dispersion interferometry based procedure.

A fourth category of embodiments of the several different categoriescomprise both an apparatus and method for measuring and compensating foreffects of cyclic errors in dispersion related signals and in bothdispersion related signals and distance measuring related signals ofdistance measuring interferometry. Dispersion interferometry is used todetermine the effects of a gas on the measured optical path of thedistance measuring interferometry and an apparatus and method is usedfor detecting and compensating for the effects of cyclic errors in thedispersion related signals and in the distance measuring relatedsignals. Effects of cyclic errors are compensated in the dispersionmeasuring related signals or the dispersion measuring and the distancemeasuring related signals using measured effects of cyclic error. Themeasured effects of cyclic error are determined from analyses of Fouriertransforms of phases of electrical interference signals wherein theelectrical interference signals are generated by detection ofpolarization mixed reference and measurement beams from the distancemeasuring and/or dispersion measuring interferometers.

Embodiments in a fifth category of the several different categoriescomprise both an apparatus and method for measuring and correcting forcyclic errors in both a dispersion measuring related signal and arefractivity measuring related signal or refractivity measuring relatedsignals used to determine intrinsic optical properties of a gas.Embodiments in the fifth category of the several different categoriesalso comprise both an apparatus and method for measuring and correctingfor cyclic errors in a wavelength measuring and/or related signal usedto determine and/or monitor the wavelength of an optical beam.

FIG. 2a depicts in schematic form an apparatus and method in accordancewith the first embodiment of the present invention. The first embodimentis from the first category of embodiments. The interferometer depictedin FIG. 2a is a polarizing, heterodyne, single pass interferometer.Although the first embodiment comprises a heterodyne system, the instantinvention is readily adapted for use in a homodyne system in which thereference and measurement beams have the same frequencies beforeintroduction of any Doppler shifts. While the apparatus has applicationfor a wide range of radiation sources, the following description istaken by way of example with respect to an optical measuring system.

Referring to FIG. 2a, a light beam 107 emitted from source 101 passesthrough a modulator 103 becoming light beam 109. Modulator 103 isexcited by a driver 105. Source 101 is preferably a laser or like sourceof coherent radiation, preferably polarized, and having a wavelength λ₂.Modulator 103 may, for example, be an acousto-optic device or acombination of acousto-optic devices with additional optics forselectively modulating polarization components of beam 107. Modulator103 preferably shifts the oscillation frequency of one linearlypolarized component of beam 107 an amount f₂ with respect to anorthogonally linearly polarized component, the directions ofpolarizations of the non-frequency and frequency shifted componentsbeing parallel and orthogonal, respectively, to the plane of FIG. 2a.The oscillation frequency f₂ is determined by the driver 105.

Light source 101 such as a laser can be any of a variety of frequencymodulation apparatus and/or lasers. For example, the laser can be a gaslaser, e.g., a HeNe laser, stabilized in any of a variety ofconventional techniques known to those skilled in the art, see forexample, T. Baer et al., “Frequency Stabilization of a 0.633 μmHe—Ne-longitudinal Zeeman Laser,” Applied Optics, 19, 3173-3177 (1980);Burgwald et al., U.S. Pat. No. 3,889,207, issued Jun. 10, 1975; andSandstrom et al., U.S. Pat. No. 3,662,279, issued May 9, 1972.Alternatively, the laser can be a diode laser frequency stabilized inone of a variety of conventional techniques known to those skilled inthe art, see for example, T. Okoshi and K. Kikuchi, “FrequencyStabilization of Semiconductor Lasers for Heterodyne-type OpticalCommunication Systems,” Electronic Letters, 16, 179-181 (1980) and S.Yamaqguchi and M. Suzuki, “Simultaneous Stabilization of the Frequencyand Power of an AlGaAs Semiconductor Laser by Use of the OptogalvanicEffect of Krypton,” IEEE J. Quantum Electronics, QE-19, 1514-1519(1983).

Two optical frequencies may be produced by one of the followingtechniques: (1) use of a Zeeman split laser, see for example, Bagley etal., U.S. Pat. No. 3,458,259, issued Jul. 29, 1969; G. Bouwhuis,“Interferometrie Mit Gaslasers,” Ned. T. Natuurk, 34, 225-232 (August1968); Bagley et al., U.S. Pat. No. 3,656,853, issued Apr. 18, 1972; andH. Matsumoto, “Recent interferometric measurements using stabilizedlasers,” Precision Engineering, 6(2), 87-94 (1984); (2) use of a pair ofacousto-optical Bragg cells, see for example, Y. Ohtsuka and K. Itoh,“Two-frequency Laser Interferometer for Small Displacement Measurementsin a Low Frequency Range,” Applied Optics, 18(2), 219-224 (1979); N.Massie et al., “Measuring Laser Flow Fields With a 64-Channel HeterodyneInterferometer,” Applied Optics, 22(14), 2141-2151 (1983); Y. Ohtsukaand M. Tsubokawa, “Dynamic Two-frequency Interferometry for SmallDisplacement Measurements,” Optics and Laser Technology, 16, 25-29(1984); H. Matsumoto, ibid.; P. Dirksen, et al., U.S. Pat. No.5,485,272, issued Jan. 16, 1996; N. A. Riza and M. M. K. Howlader,“Acousto-optic system for the generation and control of tunablelow-frequency signals,” Opt. Eng., 35(4), 920-925 (1996); (3) use of asingle acousto-optic Bragg cell, see for example, G. E. Sommargren,commonly owned U.S. Pat. No. 4,684,828, issued Aug. 4, 1987; G. E.Sommargren, commonly owned U.S. Pat. No. 4,687,958, issued Aug. 18,1987; P. Dirksen, et al., ibid.; (4) use of two longitudinal modes of arandomly polarized HeNe laser, see for example, J. B. Ferguson and R. H.Morris, “Single Mode Collapse in 6328 Å HeNe Lasers,” Applied Optics,17(18), 2924-2929 (1978); (5) use of birefringent elements or the likeinternal to the laser, see for example, V. Evtuhov and A. E. Siegman, “A“Twisted-Mode” Technique for Obtaining Axially Uniform Energy Density ina Laser Cavity,” Applied Optics, 4(1), 142-143 (1965); or the use of thesystems described in U.S. patent application with Ser. No. 09/061,928filed Apr. 17, 1998 entitled “Apparatus to Transform Two Non-ParallelPropagating Optical Beam Components into Two Orthogonally Polarized BeamComponents” and U.S. patent application with Ser. No. 09/507,529 filedFeb. 18, 2000 entitled “Apparatus for Generating Linearly-OrthogonallyPolarized Light Beams” both by Henry A. Hill, the contents of bothapplications which are incorporated herein by reference.

The specific device used for the source of beam 109 will determine thediameter and divergence of beam 109. For some sources, e.g., a diodelaser, it will likely be necessary to use conventional beam shapingoptics, e.g., a conventional microscope objective, to provide beam 109with a suitable diameter and divergence for elements that follow. Whenthe source is a HeNe laser, for example, beam shaping optics may not berequired.

As shown in FIG. 2a, interferometer 169 comprises a referenceretroreflector 191, object retroreflector 192, quarter wave phaseretardation plates 177 and 178, and a polarizing beam splitter 171. Thisconfiguration is known in the art as a polarized Michelsoninterferometer. The position of object retroreflector 192 is controlledby translator 167.

Beam 109 incident on interferometer 169 results in beams 133 and 134 asillustrated in FIG. 2a. Beams 133 and 134 contain information atwavelength λ₂ about the optical path length through the measuring path198 and about the optical path length through the reference path,respectively. Beams 133 and 134 exit interferometer 169 and enterdetector system 189 illustrated in diagrammatic form in FIG. 2a. Indetector system 189, beam 133 is reflected by mirror 163A, reflected bymirror 163B, incident on polarizing beam splitter 163C, and a portionthereof reflected by polarizing beam splitter 163C to become a firstcomponent of beam 141. Beam 134 is reflected by mirror 163A, incident ofpolarizing beam splitter 163C, and a portion thereof transmitted bypolarizing beam splitter 163C to become a second component of beam 141.

Interferometer 169 introduces phase shift φ₂ between the first andsecond components of beam 141 so that beam 141 is a phase-shifted beam.The magnitude of phase shift φ₂ is related to round-trip physical lengthL₂ of measurement path 198 according to the formulae

φ₂ =L ₂ p ₂ k ₂ n ₂  (1)

where p₂ is the number of passes through the respective reference andmeasurement legs, n₂ is the refractive index of a gas in measurementpath 198 corresponding to the optical path introducing the phase shiftφ₂ and to wavenumber k₂=2π/λ₂. The interferometer shown in FIG. 2a isfor p₂=1 so as to illustrate in the simplest manner the function of theapparatus of the first embodiment. To those skilled in the art, thegeneralization to the case when p₂≠1 is a straight forward procedure.The value for L₂ corresponds to twice the difference between thephysical length of measurement path 198 and an associated referencepath.

In a next step as shown in FIG. 2a, phase-shifted beam 141 passesthrough polarizer 179, impinges upon photodetector 185, and generates anelectrical interference signal, heterodyne signal s₂, preferably byphotoelectric detection. Polarizer 179 is oriented so as to mixpolarization components of phase-shifted beam 141. Signal s₂ may bewritten in a spectral representation of the form $\begin{matrix}{s_{2} = \quad {{a_{2,1,0,1,0}{\cos \left( {{\omega_{2}t} + \phi_{2} + \zeta_{2,1,0,1,0}} \right)}} +}} \\{\quad {{\sum\limits_{u,u^{\prime},p,p^{+}}{a_{2,u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega_{2}t} + \omega_{2u^{\prime}}^{\prime} + {p\quad \phi_{2}} - {p^{+}\phi_{2}^{+}} + \zeta_{2,u,u^{\prime},p,p^{+}}} \right)}}} +}} \\{\quad {{{\sum\limits_{q}{\left( a_{2,1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{\quad_{2,1,0,1,0,q,q}}{\cos \left\lbrack {{q\left( {{\omega_{2}t} + \phi_{2}} \right)} + \zeta_{2,1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{\quad_{2,1,0,1,0,q,{q - 2}}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega_{2}t} + \phi_{2}} \right)} + \zeta_{2,1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{2,1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega_{2}t} + \phi_{2}} \right)} + \zeta_{2,1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}} + \ldots}\quad,}}\end{matrix}$

u=0,1;

u′=0,1, . . . ; except u′≠0 if u=1, p=1, p⁺=0; ω′_(2,0)=0;

p,p⁺=0,1, . . . ; w_(2,1)/w_(2,2);

p⁺≠0 if p=1 and u=1;

w_(2,1),w_(2,2)=1,2, . . . ; w_(2,1)≠w_(2,2);

q=2,3, . . . ;

where

φ₂ ⁺ =L ₂ p ₂ k ₂ ⁺ n ₂;  (3)

k ₂ ⁺=2π[(1/λ₂)+(f ₂ /c)];  (4)

ω′_(2u′) is a set of angular frequencies not including ω; q_(R)=1 or 0depending on whether q is an odd or even integer, respectively; and c isthe speed of light in vacuum.

Terms in Eq. (2) with p⁺≧1 arise as a result of a portion of thereference beam component of the beam 109 being transmitted throughpolarizing beam splitter 171 and passing through the measurement beampath 198. The terms with u=0 may arise as a result of a portion of themeasurement beam component of beam 109 being reflected by polarizingbeam splitter 171, passing through the reference path of interferometer169, and being detected as an electrical interference signal generatedby detection of the portion of the measurement beam component and themeasurement beam component that passes through the measurement path 198.The terms with u=0 may also arise as a result of a portion of thereference beam component of beam 109 being transmitted by polarizingbeam splitter 171, passing through the measurement beam path 198, anddetected as an electrical interference signal generated by detection ofthe portion of the reference beam component of the beam 109 and thereference beam component that passes through the reference path ofinterferometer 169. The terms with u′≠0 may arise from intensityfluctuations in beam 107 that as can be produced by one or moreswitching frequencies in power supplies of source 101.

The parameter q is a nonlinearity order index where the nonlinearityarises as a result of nonlinearities in detector 185 and/or theanalogue-to-digital converter used to convert s₂ from an analogue signalto a digital format. Coefficients B_(2,u,u′,p,p) _(⁺) _(, q,m) arerelated to the coefficients in the expansion of (cos x)^(q) in terms ofcos(q−m)x, m=q,q−2, . . . , q_(R). The coefficients a_(2,u,u′,p,p) _(⁺), phase offsets ζ_(2,u,u′,p,p) _(⁺) , and coefficients B_(2,u,u′,p,p)_(⁺) _(,q,m) may be functions of system properties such as degree ofoverlap of reference and measurement beam components of an output beam,the angular frequency {dot over (φ)}₂, and the intensity of beam 109 butare otherwise substantially constant in time.

There are terms not explicitly represented in Eq. (2) that are due tohigher order effects, a higher order than those explicitly representedin Eq. (2). Effects of the higher order effect terms are typically lessthan effects of the terms explicitly represented in Eq. (2). However,should it be necessary to include for a given application any of thehigher order effect terms not explicitly represented in from Eq. (2),such terms will be identified in an initialization procedure andoperating procedure of the first embodiment and thereby included indescribed cyclic error compensation procedure of the first embodiment.

The dominant term in Eq. (2) has a phase dependence of(ω₂t+φ₂+ζ_(2,1,0,1,0)) and coefficient a_(2,1,0,1,0). The remainingterms in Eq. (2), hereinafter denoted as s_(2,ψ), i.e.

s _(2,ψ) =s ₂ −a _(2,1,0,1,0) cos(ω₂ t+φ ₂+ζ_(2,1,0,1,0)),  (5)

correspond to cyclic error terms.

Heterodyne signal s₂ is transmitted to electronic processor 127 foranalysis as electronic signal 123 in either digital or analog format,preferably in digital format. Electronic signal 123 further comprises aNyquist angular frequency ω_(2,Ny) determined by the sampling frequencyof an analog-to-digital converter, preferably in detector 185, used inthe conversion of s₂ to a digital format.

The phase of driver 105 is transmitted by electrical signal S_(2,Ref),reference signal 121, in either digital or analog format, preferably indigital format, to electronic processor 127. A reference signal, analternative reference signal to reference signal 121, may also begenerated by an optical pick off means and detector (not with anon-polarizing beam splitter, mixing the portion of the beam 109 that issplit off, and detecting the mixed portion to produce an alternativeheterodyne reference signal.

Referring to FIG. 2b, electronic processor 127 comprises electronicprocessor 151B where a Fourier transform F(s₂) of heterodyne signal s₂is generated by either digital or analog signal processes, preferably adigital process such as a finite Fourier transform algorithm (FFT).Electronic processor 127 further comprises spectrum analyzer 151A thatprocesses reference signal S_(2,Ref) for ω₂=2πf₂. Spectrum analyzer 151Ais preferably based on a sliding window Fourier transform algorithm.

In a next step, Fourier transform F(s₂) and angular frequencies ω₂ andω_(2,Ny) are transmitted to electronic processor 153 where complexspectral coefficients in F(s₂) corresponding to cyclic error terms ins_(2,ψ) are extracted at angular frequencies {tilde over (ω)}_(2,v) andaliases of {tilde over (ω)}_(2,v). An amplitude of a cyclic error termin s_(2,ψ) corresponds to the amplitude of a corresponding peak in anassociated power spectrum and the phase of the cyclic error term ins_(2,ψ) corresponds to the arctan of the ratio of the imaginary and realcomponents of F(s₂) at the angular frequency of the corresponding peakin the associated power spectrum. Angular frequencies {tilde over(ω)}_(2,v), where v is an index parameter comprising u, u′, p, p⁺, andq, correspond to the set of angular frequencies equal to derivatives,with respect to time, of the arguments of the sinusoidal factors in theterms of s_(2,ψ). Aliases {tilde over (ω)}_(2,v,A) of {tilde over(ω)}_(2,v) are given by the formula $\begin{matrix}{{{{{\overset{\sim}{\omega}}_{2,v,A} = {{\left( {- 1} \right)^{r}{\overset{\sim}{\omega}}_{2,v}} - {\left\lbrack {{\left( {- 1} \right)^{r}\left( {r + \frac{1}{2}} \right)} - \left( \frac{1}{2} \right)} \right\rbrack \omega_{2,{Ny}}}}};{r = 1}},2,\ldots}\quad} & (6)\end{matrix}$

with

rω _(2,Ny)<{tilde over (ω)}_(2,v)<(r+1)ω_(2,Ny).  (7)

In practice, the amplitudes and associated phases of cyclic error termsin s_(2,ψ) need be extracted only for a small subset of the set ofpossible {tilde over (ω)}_(2,v) and {tilde over (ω)}_(2,v,A). Theselection of the subset of the set of possible {tilde over (ω)}_(2,v)and {tilde over (ω)}_(2,v,A) may be guided by properties of certainterms in Eq. (2). However, as part of an initialization procedure, theselection of the subset of the set of possible {tilde over (ω)}_(2,v)and {tilde over (ω)}_(2,v,A) is based on a power spectrum analysis of s₂and chi-square tests of peaks in the power spectrum. The chi-squaretests identify statistically significant peaks in the power spectrum.Representation of the angular frequencies {tilde over (ω)}_(2,v) and{tilde over (ω)}_(2,v,A) of the subset of {tilde over (ω)}_(2,v) and{tilde over (ω)}_(2,v,A) associated with the statistically significantpeaks in terms of ω₂, ω_(2,Ny), {dot over (φ)}₂, {dot over (φ)}₂ ⁺, u,u′, p, p⁺, q, w_(2,1)/w_(2,2), and r is determined by observingproperties of the respective {tilde over (ω)}2,v and {tilde over(ω)}_(2,v,A) as {dot over (φ)}₂ is varied where {dot over (φ)}₂=dφ₂/dtand {dot over (φ)}₂ ⁺=dφ₂ ⁺/dt. Note that {dot over (φ)}₂ ⁺={dot over(φ)}₂ to a relative precision of the order of or less than 10⁻⁶.

The initialization procedure is performed by electronic processor 153.As part of an operating procedure of the first embodiment, powerspectrum analyses of s₂ and chi-square tests of peaks in the powerspectra are monitored for possible changes that may need be made to thesubset of the set of possible {tilde over (ω)}_(2,v) and ω_(2,v,A)during operation of the apparatus and method of the first embodiment.The power spectrum analyses of s₂ and associated chi-square testsexecuted as part of the monitoring procedure are also performed as abackground task by electronic processor 153.

Cyclic error terms in s_(2,ψ) comprise terms generated by severalmechanisms and hereinafter will be referred to broadly as comprisingcoherent cyclic errors. For some configurations of interferometers, inparticular multiple pass interferometers, it is possible for a systemcomprising a source, interferometer, and detector to generate coherentcyclic errors that comprise subharmonics of {dot over (φ)}₂. Cyclicerrors terms in s_(2,ψ) comprising subharmonics of {dot over (φ)}₂correspond to terms in Eq. (2) with p=w_(2,1)/w_(2,2),

w_(2,1), w_(2,2)=1,2, . . . , w_(2,1)≠w_(2,2) and/or p⁺=w_(2,1)/w_(2,2),

w_(2,1), w_(2,2)=1,2, . . . , w_(2,1)≠w_(2,2).

One example of subharmonic cyclic error generation can be in adifferential plane mirror interferometer where a ghost beam is generatedas a result of one reflection by an object mirror, and a secondreflection from a nominally transmissive surface of a quarter wave phaseretardation plate. When the reflecting surface of the object mirror andthe nominally transmissive surface of the quarter wave phase retardationplate are parallel, the ghost beam and reference beam components ofoutput beam have directions of propagation that are parallel. Asubsequent detection of a mixed output beam comprising ghost andreference beam components is a heterodyne signal with a subharmoniccyclic error.

Another example of subharmonic cyclic error generation can be in a highstability plane mirror interferometer, HSPMI, comprising a polarizingbeam splitter, a measurement object and reference plane mirrors, and aretroreflector. The state of polarizations of an input beam impinging ona retroreflector and a corresponding exit beam are generally different,e.g., for a linearly polarized input beam, the exit beam typically iselliptically polarized with the major axis of the ellipse rotated withrespect to the plane of polarization of the input beam (see N. Bobroff,“Recent advances in displacement measuring interferometry,” Measurementand Sci. & Tech., 4(9), 907-926, 1993). The ellipticity of the exit beamgenerates measurement and reference beam components in the HSPMI outputbeam that have made only a single pass instead a double pass to themeasurement object mirror. The single pass components when mixed withother components of the HSPMI output beam produce subharmonic cyclicerrors in subsequently generated interference signal.

In addition, it is possible for a system comprising a source,interferometer, detector, and digital signal processing to generatecoherent cyclic errors that comprise neither subharmonics or harmonicsof {dot over (φ)}₂ but are related to harmonics of ω₂+{dot over (φ)}₂,ω₂, and other angular frequencies. Coherent cyclic errors that haveangular frequencies that are not subharmonics, not harmonics ofsubharmonics, and not harmonics of {dot over (φ)}₂ can be produced forexample by nonlinearities in the detector and/or amplifiers generatings₂ and in an analog-to-digital converter used to digitize s₂ and haveangular frequencies which are harmonics of ω₂+{dot over (φ)}₂, ω₂, andother angular frequencies. Coherent cyclic errors that have angularfrequencies that are not subharmonics, not harmonics of subharmonics,and not harmonics of {dot over (φ)}₂ can also be produced for example byaliasing in digital signal processing and have angular frequencies whichare aliases of subharmonics and harmonics of {dot over (φ)}₂ and aliasesof ω₂+{dot over (φ)}₂, ω₂, and other angular frequencies and harmonicsthereof. The aliases are related to the Nyquist angular frequencyω_(2,Ny).

An example of the various coherent cyclic errors that are harmonics of{dot over (φ)}₂ and coherent cyclic errors that have angular frequenciesthat are not subharmonics and not harmonics of {dot over (φ)}₂ is showngraphically in FIG. 2d as a function of {dot over (φ)}₂. Coherent cyclicerrors that are harmonics and subharmonics of {dot over (φ)}₂ areindicated in FIG. 2d as solid lines 90, 91, and 92, for example, withline 91 being the first harmonic. Coherent cyclic errors that are theresult for example of nonlinearities in the detector and amplifiersgenerating s₂ and in analog-to-digital converters are indicated in FIG.2d as solid line 95. Coherent cyclic errors that are the result ofaliasing are indicated in FIG. 2d as dashed lines that reflect from line99 representing the Nyquist frequency.

The amplitudes and phase offsets of terms in the spectral representationgiven by Eq. (2) will in general depend on the magnitude of the rate ofchange of a phase associated with the term as a result, for example, ofproperties of group delay experienced by the heterodyne signal. Groupdelay, often called envelope delay, describes the delay of a packet offrequencies and the group delay at a particular frequency is defined asthe negative of the slope of the phase curve at the particular frequency[see H. J. Blinchikoff and A. I. Zverev, Filtering in the Time andFrequency Domains, Section 2.6, 1976 (Wiley, New York)].

The extracted complex spectral coefficients in F(s₂) corresponding tocyclic error terms in s_(2,ψ) are then sent to electronic processor 154where the extracted spectral coefficients are normalized, filtered withrespect to time, and interpolations made as required and amultidimensional array of normalized, filtered, and interpolated complexspectral coefficients maintained. The step of normalization is for thepurpose of compensating for effects of non-zero values of second andhigher order derivatives of φ₂ with respect to time that exist at thetime of a determination of the set of complex spectral coefficients. Thedimensionality of the multidimensional array is determined in part bythe magnitude of the filtered complex spectral coefficients, therequired precision of an end use application with respect to correctionfor coherent cyclic errors, and the dependence of the filtered complexspectral coefficients on {dot over (φ)}₂ and other system properties.

For the next step in electronic processor 127 as shown in FIG. 2b,electronic procession 155 computes the coherent cyclic error corrections_(2,ψ,M) using information listed in the multidimensional array ofnormalized, filtered, and interpolated complex spectral coefficients andgenerated by electronic processor 154. Electronic processor 152 computess₂−s_(2,ψ,M) to compensate for coherent cyclic errors in s₂. Coherentcyclic error compensated signal s₂−s_(2,ψ,M) and angular frequency ω₂are transmitted to electronic processor 256 where the phase ofs₂−s_(2,ψ,M), φ₂, is determined by a phase detector such as a slidingwindow FFT, a zero crossing phase detector, or the like. Phase φ₂ istransmitted as signal 128 to digital computer 129 for use in downstreamapplications such as determining linear displacements of object 192 noteffected by coherent cyclic errors.

Each of the electronic processors comprising electronic processor 127preferably performs respective functions as digital processes.

The formalism for the normalization of extracted complex spectralcoefficients is next described. The Fourier transform of s₂ comprisesFourier transforms of terms having factors such as

cos ζ_(2,u,u′,p,p) _(⁺) cos(uω₂t+ω′_(u′)+pφ₂−p⁺φ₂ ⁺) and

sin ζ_(2,u,u′,p,p) _(⁺) sin(uω₂t+ω′_(u′)+pφ₂−p⁺φ₂ ⁺) as evident from Eq.(2). The Fourier transform of a sinusoidal function sin β is related tothe Fourier transform of cos β as

F(sin β)=F{cos[β−(π/2)]}  (8)

where β is a function of time and representive of arguments ofsinusoidal factors in Eq. (2). For evaluation of the Fourier transformof cos β, factor cos β is written as

 cosβ=cos[β(T)+{dot over (β)}(T)(t−T)]cos[β(t)−β(T)−{dot over(β)}(T)(t−T)]−sin[β(T)+{dot over (β)}(T)(t−T)]sin[β(t)−β(T)−{dot over(β)}(T)(t−T)]  (9)

and factors cos[β(t)−β(T)−{dot over (β)}(T)(t−T)] and sin[β(t)−β(T)−{dotover (β)}(T)(t−T)] in Eq. (9) are expanded in Taylor's series about t=Twhere {dot over (β)}(T)=[dβ/dt]_(t=T). The Taylor's series expansionsincluding terms up through fifth order in (t−T) may be expressedaccording to the formulae $\begin{matrix}\begin{matrix}{{\cos \left\lbrack {{\beta (t)} - {\beta (T)} - {{\overset{.}{\beta}(T)}\left( {t - T} \right)}} \right\rbrack} = \quad {1 - {{3\left\lbrack {\overset{..}{\beta}(T)} \right\rbrack}^{2}\quad \frac{\left( {t - T} \right)^{4}}{4!}}}} \\{\quad {{{10\quad {\overset{..}{\beta}(T)}{\overset{...}{\beta}(T)}\quad \frac{\left( {t - T} \right)^{5}}{5!}} + \ldots}\quad,}}\end{matrix} & (10) \\\begin{matrix}{{\sin \left\lbrack {{\beta (t)} - {\beta (T)} - {{\overset{.}{\beta}(T)}\left( {t - T} \right)}} \right\rbrack} = \quad {{{\overset{..}{\beta}(T)}\quad \frac{\left( {t - T} \right)^{2}}{2!}} +}} \\{\quad {{{\overset{...}{\beta}(T)}\quad \frac{\left( {t - T} \right)^{3}}{3!}} +}} \\{\quad {{{\overset{IV}{\beta}(T)}\quad \frac{\left( {t - T} \right)^{4}}{4!}} +}} \\{\quad {{{{\overset{V}{\beta}(T)}\quad \frac{\left( {t - T} \right)^{5}}{5!}} + \ldots}\quad,}}\end{matrix} & (11) \\{{where}\text{}{{{\overset{..}{\beta}(T)} = \left( \left\lbrack {{^{2}\beta}/{t^{2}}} \right\rbrack \right)_{t = T}},{{\overset{...}{\beta}(T)} = \left( \left\lbrack {{^{3}\beta}/{t^{3}}} \right\rbrack \right)_{t = T}},{{\overset{IV}{\beta}(T)} = \left( \left\lbrack {{^{4}\beta}/{t^{4}}} \right\rbrack \right)_{t = T}},\quad {{{and}\quad {\overset{V}{\beta}(T)}} = {\left( \left\lbrack {{^{5}\beta}/{t^{5}}} \right\rbrack \right)_{t = T}.}}}} & \quad\end{matrix}$

For a given term in s_(2,ψ), the corresponding second and higher orderderivatives of β with respect to time are proportional to second andhigher order derivatives of φ₂ with respect to time, respectively, witha proportionally constant determined by properties of the respectivesinusoidal factor in the given term. The proportionally constant may bezero or non-zero. The Fourier transform of cos β over time intervalT−τ/2 to T+τ/2 can be expressed, using representations in Eqs. (9),(10), and (11), as $\begin{matrix}{{{F\left( {\cos \quad \beta} \right)} = {\tau \quad \frac{^{\quad \omega \quad T}}{2\sqrt{2\quad \pi}}\quad \left( \begin{pmatrix}{{^{\quad \beta \quad {(T)}}C^{+}\left\{ \frac{\left\lbrack {\omega + {\overset{.}{\beta}(T)}} \right\rbrack \tau}{2} \right\}} +} \\{^{{- }\quad \beta \quad {(T)}}C^{-}\left\{ \frac{\left\lbrack {\omega - {\overset{.}{\beta}(T)}} \right\rbrack \tau}{2} \right\}}\end{pmatrix} \right)}},} & (12)\end{matrix}$

where $\begin{matrix}\begin{matrix}{{C^{+}(x)} = \quad {{j_{0}\left( \frac{x}{2} \right)} - {{g_{2}\left( \frac{x}{2} \right)}\left( \frac{1}{2!} \right)\left( \frac{\tau}{2} \right)^{2}{\overset{..}{\beta}(T)}} -}} \\{\quad {{{g_{3}\left( \frac{x}{2} \right)}\left( \frac{1}{3!} \right)\left( \frac{\tau}{2} \right)^{3}{\overset{...}{\beta}(T)}} -}} \\{\quad {{{g_{4}\left( \frac{x}{2} \right)}\frac{1}{4!}\left( \frac{\tau}{2} \right)^{4}\left\{ {{\overset{IV}{\beta}(T)} + {3\left\lbrack {\overset{..}{\beta}(T)} \right\rbrack}^{2}} \right\}} -}} \\{\quad {{{{g_{5}\left( \frac{x}{2} \right)}\frac{1}{5!}\left( \frac{\tau}{2} \right)^{5}\left\{ {{\overset{V}{\beta}(T)} + {10{\overset{..}{\beta}(T)}{\overset{...}{\beta}(T)}}} \right\}} + \ldots}\quad,}}\end{matrix} & (13) \\\begin{matrix}{{C^{-}(x)} = \quad {{j_{0}\left( \frac{x}{2} \right)} + {{g_{2}\left( \frac{x}{2} \right)}\left( \frac{1}{2!} \right)\left( \frac{\tau}{2} \right)^{2}{\overset{..}{\beta}(T)}} +}} \\{\quad {{{g_{3}\left( \frac{x}{2} \right)}\left( \frac{1}{3!} \right)\left( \frac{\tau}{2} \right)^{3}{\overset{...}{\beta}(T)}} +}} \\{\quad {{{g_{4}\left( \frac{x}{2} \right)}\frac{1}{4!}\left( \frac{\tau}{2} \right)^{4}\left\{ {{\overset{IV}{\beta}(T)} - {3\left\lbrack {\overset{..}{\beta}(T)} \right\rbrack}^{2}} \right\}} +}} \\{\quad {{{{g_{5}\left( \frac{x}{2} \right)}\frac{1}{5!}\left( \frac{\tau}{2} \right)^{5}\left\{ {{\overset{V}{\beta}(T)} - {10{\overset{..}{\beta}(T)}{\overset{...}{\beta}(T)}}} \right\}} + \ldots}\quad,}}\end{matrix} & (14) \\{{{g_{1}(x)} = {\frac{i}{2}\quad {j_{1}(x)}}},} & (15) \\{{g_{2}(x)} = {\frac{1}{3}\quad\left\lbrack {{j_{0}(x)} - {2\quad {j_{2}(x)}}} \right\rbrack}} & (16) \\{{{g_{3}(x)} = {\frac{i}{5}\quad\left\lbrack {{3{j_{1}(x)}} - {2\quad {j_{3}(x)}}} \right\rbrack}},} & (17) \\{{{g_{4}(x)} = {\frac{1}{35}\quad\left\lbrack {{7{j_{0}(x)}} - {2\quad 0{j_{2}(x)}} + {8{j_{4}(x)}}} \right\rbrack}},} & (18) \\{{{g_{5}(x)} = {\frac{i}{63}\quad\left\lbrack {{27{j_{1}(x)}} - {2\quad 8{j_{3}(x)}} + {8{j_{5}(x)}}} \right\rbrack}},} & (19)\end{matrix}$

and j_(n)(x), n=0,±1,±2, . . . , is the spherical Bessel function of thefirst kind and order n [see Chapter 10 of Handbook of MathematicalFunctions, Eds. M. Abramowitz and I. Stegun, Nat. Bureau of StandardsApplied Mathematics Series 55]. The coefficients for g_(n)(x) in termsof j_(m)(x) are the same as the coefficients for x^(n) in terms ofP_(m)(x), the Legendre polynominal of degree m, multiplied by (i)^(m)[see Eq. 10.1.14 and Table 22.9 in Abramowitz and Stegun, ibid.].

In electronic processor 153, the complex spectral coefficients ofcoherent cyclic errors are extracted using the complex values of theFourier transform F(s₂) at the subset of frequencies of {tilde over(ω)}_(2,v) and {tilde over (ω)}_(2,v,A) and normalized for the effectsof non-zero second and higher order time derivatives of φ₂. Thenormalizations for non-zero second and higher order derivatives of β areobtained using Eqs. (8), (12), (13), and (14). Values for the second andhigher order time derivatives of φ₂ are obtained form the output ofelectronic processor 156 in an iterative procedure. The number of higherorder derivatives of φ₂ that must be included in making the correctionsfor non-zero second and higher order time derivatives of φ₂ isdetermined in part by the magnitude of τ and in part by the magnitudesof the second and higher order time derivatives of φ₂.

There is a set of values of {dot over (φ)}₂ for which certain of thesubset of corresponding frequencies {tilde over (ω)}_(2,v) and {dot over(ω)}_(2,v,A) and the frequency of the dominant complex peak in F(s₂)comprise two or more frequency values that are separated by less than orof the order of the angular frequency resolution of the Fouriertransform F(s₂). For the set of values of {dot over (φ)}₂, therespective values of F(s₂) represent superimposed values of respectiveFourier transforms of coherent cyclic errors and the dominant complexpeak in F(s₂) and are not included in the values of complex spectralcoefficients determined by electronic processor 153. The values ofcomplex spectral coefficients determined by electronic processor 153 aretransmitted to electronic processor 154.

Electronic processor 154 filters with respect to time the values of thenormalized complex spectral coefficients received from electronicprocessor 153 and interpolates the filtered, normalized values todetermine complex spectral coefficients of the coherent cyclic errorsnot included in the values of filtered, normalized complex spectralcoefficients determined by electronic processor 153. If the normalized,filtered, and interpolated values of the complex spectral coefficientsare determined to be dependent on {dot over (φ)}₂, the {dot over (φ)}₂dependence of the normalized, filtered, and interpolated values of thecomplex spectral coefficients may effectively be represented in a powerseries, orthogonal functions, or orthogonal polynomials of {dot over(φ)}₂.

The determined the set of normalized, filtered, and interpolated valuesof the complex spectral coefficients are transmitted to electronicprocessor 155 wherein a computed value of s_(2,ψ), s_(2,ψ,M), isgenerated.

There is a type of coherent cyclic error with a corresponding coherentcyclic error amplitude and phase offset that depends on an orientationof a measurement object mirror. This type of coherent cyclic error willhereinafter be referred to as a variable coefficient type of coherentcyclic error. For those end use applications where tilt and yaw of themeasurement object mirror are measured and monitored to a requisiteaccuracy, e.g., measured by interferometric means, the variablecoefficient type of coherent cyclic error can be measured andcompensated within the framework of the present invention.

The cyclic error complex amplitude in the coherent cyclic errorrepresentation for a variable coefficient type of coherent cyclic errorwill comprise a complex multiplicative factor. The variable type ofcoherent cyclic error will subsequently be described in terms of twosubtypes. The complex multiplicative factor for a first subtype of thetwo subtypes will generally be function of the degree of overlap of aghost beam and a non-ghost beam components or of two ghost beamcomponents of an output beam and the function can be modeled and/ormeasured experimentally. For the first subtype, the wavefronts ofcorresponding ghost beam and non-ghost beam components or two ghost beamcomponents of the output beam are parallel independent of theorientation of the object mirror. The complex multiplicative factor of asecond subtype of the two subtypes will generally be a function of thedegree of overlap of a ghost beam and a non-ghost beam components or oftwo ghost beam components of an output beam and the wedge angle betweenthe wavefronts of the corresponding ghost beam and the non-ghost beamcomponents or of the two corresponding ghost beam components of theoutput beam. For the second subtype, the wavefronts of the correspondingghost beam and the non-ghost beam components or of the two correspondingghost beam components of the output beam will be substantially parallelfor a particular orientation of the measurement object mirror. Theproperties of the complex multiplicative factor will be analogous toproperties of a fringe visibility function and can be modeled and/ormeasured experimentally.

For those end use applications where the variable coefficient type ofcoherent cyclic error can be eliminated by tilting of certain opticalelements or portions thereof, information on the tilt and yaw can beused to ascertain when the variable coefficient type of coherent cyclicerror is sufficiently small as to be negligible.

Complex amplitudes in a spectral representation of coherent cyclicerrors in s_(2,ψ) will also depend on the magnitude of s₂. Thedependence of the complex amplitudes on the magnitude of s₂ can bemeasured and compensated within the framework of the present invention.The complex amplitudes in the coherent cyclic error spectralrepresentation will comprise complex multiplicative factors. The complexmultiplicative factor for a specific coherent cyclic error spectralcomponent will be a function of s₂ and easily represented as a powerseries in s₂. The power series representation will be determined frommeasured properties of the respective specific complex amplitude as themagnitude of s₂ is varied. For certain interferometer and detectorsystems, the power series representation can be determined from acorresponding value of q associated with the corresponding value of{tilde over (ω)}_(2,v) or {tilde over (ω)}_(2,v,A).

One of the more subtle types of coherent cyclic errors to identify andcompensate has a {tilde over (ω)}_(2,v) that is the same as the angularfrequency of the dominant peak in |F(s₂)|² independent of the value of{dot over (φ)}₂. This type of coherent cyclic error will be referred tohereinafter as a zero-frequency-shift coherent cyclic error. Sources ofzero-frequency-shift coherent cyclic errors generally comprisenonlinearities with odd order values of nonlinearity index q greaterthan 1, e.g. q=3,5, . . . .

The primary zero-frequency-shift coherent cyclic errors arise in Eq. (2)from terms with u=1, u′=0, p=1, p⁺=0, and q_(R)=1 and can be expressedas $\begin{matrix}{\sum\limits_{{q = 3},5,\ldots}\quad \left\{ {\left( a_{2,1,0,1,0} \right)^{q}B_{2,1,0,1,0,q,1}{\cos \left\lbrack {\left( {{\omega_{2}t} + \phi_{2}} \right) + \zeta_{2,1,0,1,0,q,1}} \right\rbrack}} \right\}} & (20)\end{matrix}$

Other zero-frequency-shift coherent cyclic errors not represented in Eq.(20) arise in Eq. (2) for different values and combination of values ofu, p, and p⁺. The terms omitted in Eq. (20) arise from nonlinearcoupling between the dominant term with amplitude a_(2,1,0,1,0) andother cyclic error terms and therefore are generally several orders ofmagnitude smaller. It is evident from Eq. (20) that thezero-frequency-shift coherent cyclic errors modify the amplitude andphase offset of the measured spectral component of s₂ having a phase of(ω₂t+φ₂) with the modifications depending on properties of coherentcyclic errors present.

Thus the effects of zero-frequency-shift coherent cyclic errors aremanifested in s₂ not as cyclic errors but as errors that modify a phaseoffset at the frequency of the dominant peak in |F(s₂)|² over a range ofoperating conditions for the apparatus of the first embodiment. Inparticular, the zero-frequency-shift coherent cyclic errors alter thegroup delay properties of s₂ at the frequency of the dominant peak in|F(s₂)|². It is therefore evident that the effect of thezero-frequency-shift coherent cyclic errors are compensated in the firstembodiment for the case when the effects of group delay at the frequencyof the dominant peak in |F(s₂)|² are measured for a system comprising aninterferometer and associated electric signal processors. Preferably,the measurement is performed with the measurement leg of theinterferometer in vacuum to prevent the affects of air turbulence on thecharacterization of {dot over (φ)}.

For the case where the zero-frequency-shift coherent cyclic error termsin F(s₂) are not compensated as part of a procedure for compensation forgroup delay effects, two procedures are described in the followingparagraphs for isolating and detecting effects of thezero-frequency-shift coherent cyclic errors.

Characterization of the zero-frequency-shift errors can be importantbecause the phase offset they produce in the dominant peak of |F(s₂)|²will vary with the intensity of the measurement and reference beams andthe relative overlap of the beams that produce the optical interferencesignal—parameters that may change during operation of the interferometrysystem. Furthermore, the phase offset may additionally vary with theDoppler shift.

The first of the two procedures described for isolating and detectingeffects of the zero-frequency-shift coherent cyclic errors is thesimpler of the two procedures to implement although of a restricteddomain of applicability. For each coherent cyclic error of the set ofzero-frequency-shift coherent cyclic errors, there are correspondingcoherent cyclic errors with values of {tilde over (ω)}_(2,v) that aredifferent from the angular frequencies of the dominant peak in |F(s₂)|².For example, for an odd order nonlinearity order index of q=n≧3, the setof different {tilde over (ω)}_(2,v) may be written as 3ω_(d),5ω_(d), . .. ,nω_(d) where ω_(d) is the angular frequency of the dominant complexpeak in F(s₂). For a F(s₂) where the corresponding coherent cyclicerrors correspond to only one value of n, ratios of coefficients of thezero-frequency-shift coherent cyclic error and of the correspondingcoefficients of the zero-frequency-shift coherent cyclic errors arespecified for certain apparatus by coefficients comprising ratios ofB_(q,n) for respective values of n. Electronic processor 154 examinesthe set of corresponding coherent cyclic errors coefficients anddetermines the value of the zero-frequency-shift coherent cyclic errorcoefficient using the ratios between the coefficients of thezero-frequency-shift coherent cyclic errors and of the set ofcorresponding coherent cyclic error coefficients. For a case where morethe one value of q is required in the description of the correspondingcoherent cyclic errors, the corresponding zero-frequency-shift coherentcyclic error coefficients can be determined from a least squaresanalysis of set of corresponding coherent cyclic error coefficients, aset of corresponding simultaneous equations, and a set of correspondingbinomial coefficients.

Effectiveness of the first procedure for isolating and detecting effectsof the zero-frequency-shift coherent cyclic errors is diminished whenthe set of different {tilde over (ω)}_(2,v) include angular frequencieswhere the transfer function of the analog portion of detector 185generating s₂ is not substantially constant.

The second procedure described for isolating and detecting effects ofthe zero-frequency-shift coherent cyclic errors is based on the use oftwo input beams 109 and 109T (beam 109T is not shown in FIG. 2a). Beam109T is similar to beam 109 in that it has two orthogonally polarizedcomponents having a frequency splitting ω_(2,T). However, the frequencysplitting ω_(2,T) between the components of beam 109T is selected to bedifferent from the frequency splitting ω₂ between the components of beam109. The absolute value of the difference between the frequencysplittings, |ω_(2,T)−ω₂|, is selected to be less than both ω₂+{dot over(φ)} and ω_(2,Ny). Beam 109T propagates through interferometer 169 anddetector 189 to generate an interference signal component s_(2,T) insignal 123. The description of s_(2,T) is the same as correspondingportions of the description given for s₂. The properties of theapparatus of the first embodiment that generate the zero-frequency-shiftcoherent cyclic errors also generate a closely spaced multiplet withrespect to frequency in the power spectrum of s₂+s_(2T) when beam 109Tis present. The frequency spacing between contiguous members of themultiplet is equal to the frequency separation of beams 109 and 109T.The amplitude and phase of the zero-frequency-shift coherent cyclicerrors can be expressed by a relationship in terms of amplitudes andphases of members of the multiplet. In the procedure to isolate thezero-frequency-shift coherent cyclic errors, the primaryzero-frequency-shift coherent cyclic errors are determined using therelationship and measured amplitudes and phases of the members of themultiplet.

The description of the source of beam 109T (source of beam 109T is notshown in FIG. 2a) is the same as corresponding portions of thedescription given for the source of beam 109. For example, as shown inFIG. 2e, the interferometry system can include a first source for beam109 including light source 101 and modulator 103 excited by driver 105,a second source for beam 109T including another light source 101T andanother modulator 103T excited by another driver 105T, and optics 106 tocombine the beams produced by the first and second sources. In anotherembodiment shown in FIG. 2f, beams 109 and 109T are both derived fromthe same light source, laser 101′. For example, laser 101′ can producetwo beams 108 and 108T corresponding to different longitudinal modes oflaser 101′. Modulators 103 and 103T excited by drivers 105 and 105T,respectively, then act on beams 108 and 108T to produce input beams 109and 109T. In further embodiments, a single modulator driven at bothsplit frequencies, ω₂ and ω_(2,T), can be used. As necessary, theintensity of beams 109 and 109T may be adjusted as may the frequencysplittings. For example, the source for beam 109T can remove it from theinterferometry system once the zero-frequency-shift cyclic errors arecharacterized.

Generally, the difference between the average frequency of thecomponents of beam 109 and the average frequency of the components ofbeam 109T is selected to be greater than the Nyquist frequency ω_(2,Ny),e.g., twice as great as the Nyquist frequency ω_(2,Ny), to minimizeterms in the signal generated by detector 189 corresponding tointerference between beams 109 and 109T other than in thezero-frequency-shift multiplet. Conversely, the frequency splittingsthemselves are selected to be smaller than the Nyquist frequency.Generarally, the absolute value of the difference between the frequencysplittings |ω_(2,T)−ω₂| is selected to be smaller than either of thefrequency splittings themselves to minimize spectral overlap of themultiplet with other cyclic errors and to minimize any differencebetween the zero-shift-frequency cyclic error coefficients in s₂ ands_(2,T). For example, |ω_(2,T)−ω₂| can be selected so that|ω_(2,T)−ω₂|<(ω₂/100)

The zero-frequency-shift cyclic errors are determined from a Fourieranalyses of terms in s₂+s_(2T) such as [a_(v) cos(ω₂t+φ₂+ζ_(v))+a_(vT)cos(ω_(2T)t+φ_(2T)+ζ_(vT))]^(q) where v=(2,1,0,1,0) and q=3,5, . . . .Results from such an analysis are listed for examples of q=3,5, and 7.Results for other values of q, as required in a particular end useapplication, can be generated by the same procedure. For q=3, theprimary terms contributing to the multiplet are $\begin{matrix}{\left( a_{v} \right)^{3}B_{{v3},1}\begin{Bmatrix}\begin{matrix}\begin{matrix}{{x\quad {\cos \left\lbrack {\alpha_{2,3} - \left( {\alpha_{2,{3T}} - \alpha_{2,3}} \right)} \right\rbrack}} +} \\{{\left( {1 + {2x^{2}}} \right)\cos \quad \alpha_{2,3}} +}\end{matrix} \\{{{x\left( {2 + x^{2}} \right)}{\cos \left\lbrack {\alpha_{2,3} + \left( {\alpha_{2,{3T}} - \alpha_{2,3}} \right)} \right\rbrack}} +}\end{matrix} \\{x^{2}{\cos \left\lbrack {\alpha_{2,3} + {2\left( {\alpha_{2,{3T}} - \alpha_{2,3}} \right)}} \right\rbrack}}\end{Bmatrix}} & (21)\end{matrix}$

where $\begin{matrix}{{x = \frac{a_{vT}}{a_{v}}},} & (22)\end{matrix}$

 α_(2q)=ω₂ t+φ ₂+ζ_(vq,1),  (23)

α_(2qT)=ω_(2T) t+φ _(2T)+ζ_(vq,1T),  (24)

α_(2qT)−α_(2q) =L ₂ p ₂ n ₂(k _(2T) −k ₂)+(ζ_(2qT)−ζ_(2q)),  (25)

Note the quantity (k_(2T)−k₂) is simply 2π/c times the difference infrequency of beams 109 and 109T.

For q=5, the primary terms contributing to the multiplet are$\begin{matrix}{\left( a_{v} \right)^{5}B_{{v5},1}\begin{Bmatrix}{{x^{2}{\cos \left\lbrack {\alpha_{2,5} - {2\left( {\alpha_{2,{5T}} - \alpha_{2,5}} \right)}} \right\rbrack}} +} \\{{{x\left( {2 + {3x^{2}}} \right)}{\cos \left\lbrack {\alpha_{2,5} - \left( {\alpha_{2,{5T}} - \alpha_{2,5}} \right)} \right\rbrack}} +} \\{{\left( {1 + {6x^{2}} + {3x^{4}}} \right)\cos \quad \alpha_{2,5}} +} \\{{{x\left( {3 + {6x^{2}} + x^{4}} \right)}{\cos \left\lbrack {\alpha_{2,5} + \left( {\alpha_{2,{5T}} - \alpha_{2,5}} \right)} \right\rbrack}} +} \\{{{x^{2}\left( {3 + {2x^{2}}} \right)}{\cos \left\lbrack {\alpha_{2,5} + {2\left( {\alpha_{2,{5T}} - \alpha_{2,5}} \right)}} \right\rbrack}} +} \\{x^{3}{\cos \left\lbrack {\alpha_{2,5} + {3\left( {\alpha_{2,{5T}} - \alpha_{2,5}} \right)}} \right\rbrack}}\end{Bmatrix}} & (26)\end{matrix}$

For q=7, the primary terms contributing to the multiplet are$\begin{matrix}{\left( a_{v} \right)^{7}B_{{v7},1}{\begin{Bmatrix}{{x^{3}{\cos \left\lbrack {\alpha_{2,7} - {3\left( {\alpha_{2,{7T}} - \alpha_{2,7}} \right)}} \right\rbrack}} +} \\{{{x^{2}\left( {3 + {4x^{2}}} \right)}{\cos \left\lbrack {\alpha_{2,7} - {2\left( {\alpha_{2,{7T}} - \alpha_{2,7}} \right)}} \right\rbrack}} +} \\{{3{x\left( {1 + {4x^{2}} + {2x^{4}}} \right)}{\cos \left\lbrack {\alpha_{2,7} - \left( {\alpha_{2,{7T}} - \alpha_{2,7}} \right)} \right\rbrack}} +} \\{{\left( {1 + {12x^{2}} + {18x^{4}} + {4x^{6}}} \right)\cos \quad \alpha_{2,7}} +} \\{{{x\left( {4 + {18x^{2}} + {12x^{4}} + x^{6}} \right)}\left\lbrack {\alpha_{2,7} + \left( {\alpha_{2,{7T}} - \alpha_{2,7}} \right)} \right\rbrack} +} \\{{3{x^{2}\left( {2 + {4x^{2}} + x^{4}} \right)}{\cos \left\lbrack {\alpha_{2,7} + {2\left( {\alpha_{2,{7T}} - \alpha_{2,7}} \right)}} \right\rbrack}} +} \\{{{x^{3}\left( {4 + {3x^{2}}} \right)}{\cos \left\lbrack {\alpha_{2,7} + {3\left( {\alpha_{2,{7T}} - \alpha_{2,7}} \right)}} \right\rbrack}} +} \\{x^{4}{\cos \left\lbrack {\alpha_{2,7} + {4\left( {\alpha_{2,{7T}} + \alpha_{2,7}} \right)}} \right\rbrack}}\end{Bmatrix}.}} & (27)\end{matrix}$

Since the magnitude of the angular frequency separation is chosen to bemuch less than both ω₂+{dot over (φ)} and ω_(2,Ny), the coefficientB_(vq,1) is used to a good approximation for both the representation ofcorresponding terms in s₂ and s_(2T).

The terms in Eqs. (21), (26), and (27) are arranged according to therespective locations in respective multiplets. Note that the largestterms of the zero-frequency-shift coherent cyclic error in s₂ is a sumof the coefficients (a_(v))³ B_(v3,1) cos α_(2,3), (a_(v))⁵ B_(v5,1) cosα_(2,5), (a_(v))⁷ B_(v7,1) cos α_(2,7), . . . . Further, note that thephase difference (α_(2qT)−α_(2q)) for multiplet q can be obtaineddirectly from the difference of the measured phases of the two termscorresponding to the lowest and highest frequency components of therespective multiplet, that difference being equal to q(α_(2qT)−α_(2q)).

Continuing with the description of the procedure for isolating anddetecting effects of the zero-frequency-shift coherent cyclic errors,electronic processor 154 examines the normalized, filtered, andinterpolated values of the complex spectral coefficients for the set ofcomplex spectral coefficients corresponding to the terms in themultiplet. The zero-frequency-shift coherent cyclic error in s₂, the sumof the coefficients (a_(v))³ B_(v3,1) cos α_(2,3), (a_(v))⁵ B_(v5,1) cosα_(2,5), (a_(v))⁷ B_(v7,1) cos α_(2,7), . . . , is determined from aleast squares analysis of the set of complex spectral coefficients usingequations describing terms contributing to the multiplet such as foundin Eqs. (21), (26), and (27). The number of terms in the sum of thecoefficients (a_(v))³ B_(v3,1) cos α_(2,3), (a_(v))⁵ B_(v5,1) cosα_(2,5), (a_(v))⁷ B_(v7,1) cos α_(2,7), . . . that need to be includedis determined by the magnitudes of the terms and a required accuracy inthe compensation for the zero-frequency-shift coherent cyclic error ins₂.

Electronic processor 154 as previously cited filters the values of thenormalized complex spectral coefficients received from electronicprocessor 153 and interpolates the normalized, filtered values todetermine values of the coherent cyclic errors not included in thevalues of complex spectral coefficients determined by electronicprocessor 153. If the normalized, filtered, and interpolated values ofthe complex spectral coefficients are determined to be dependent on {dotover (φ)}₂, the {dot over (φ)}₂ dependence of the normalized, filtered,and interpolated values of the complex spectral coefficients mayeffectively be represented in a power series, a series of orthogonalfunctions, or a series of orthogonal polynomials in {dot over (φ)}₂. Theset of zero-frequency-shift coherent cyclic errors, if determined, andthe determined values of the normalized, filtered, and interpolatedvalues of the complex spectral coefficients are transmitted toelectronic processor 155 wherein a simulated value of s_(2,ψ),s^(2,ψ,M), is generated. Simulated valued s_(2,ψ,M) and s₂ aretransmitted to electronic processor 152 where s_(2,ψ,M) is subtractedfrom s₂ to yield s₂−s_(2,ψ,M). Signal s₂−s_(2,ψ,M) and frequency ω₂ aretransmitted to electronic processor 156 where phase φ₂ is determinedpreferably by a sliding window FFT or other like phase detector. Phaseφ₂ is transmitted to computer 129 as signal 128 for use in determininglinear displacements of object 192 not effected by coherent cyclicerrors.

It was noted in the description of the first embodiment that thedifferent contributions to the cyclic errors were generally the resultof imperfections in the source of beam 109, in the components ofinterferometer 169, in detector system 189, and/or in electronicprocessor 127. It is also evident from the description that propertiesof a cyclic error term with respect to frequency and amplitude generallypermit identification of the origin of the cyclic error term, i.e. withrespect to a particular type of imperfection. For example, cyclic errorsassociated with the set of frequencies ω′_(2u′) indicate an intensityfluctuation in beam 109 that could be due to an imperfection in theacousto-optic modulator 103, in the power supply system of source 101,and/or an instability in the laser. A cyclic error term associated withthe frequency {dot over (φ)} could be due to polarization mixing in beam109 and/or due to an imperfection in the polarizing beam splitterinterface 171. A cyclic error term associated with the parameter qindicates a nonlinearity in detector system 189 and/or electronicprocessor 127.

Consequently, degradation in performance of one or more components of aninterferometer system can be detected by monitoring properties of thecyclic error terms as described above with reference to FIG. 1. Suchdetection of degradation generally amounts to an early detection system,i.e. an opportunity is afforded for implementation of correctivemeasures that can generally be implemented before the interferometersystem is used in a mode unacceptable for a given end use application.The corrective measures can, for example, be part of a programmedmaintenance to correct for a degraded component in a problem areaindicated by the properties of the associated cyclic error term.Furthermore, indication of the problem area by properties of theassociated cyclic error term can substantially improve the efficiency ofexecuting the corrective measures.

The description of the first embodiment noted that the configuration ofinterferometer illustrated in FIG. 2a is known in the art as polarizedMichelson interferometer. Other forms of the Michelson interferometerand forms of other interferometers such as the high stability planemirror interferometer, or the angle-compensating interferometer orsimilar device such as is described in an article entitled “Differentialinterferometer arrangements for distance and angle measurements:Principles, advantages and applications”) by C. Zanoni, VDI Berichte Nr.749, 93-106 (1989), may be incorporated into the apparatus of thepresent invention, the foregoing article being herein incorporated byreference, as when working with stages commonly encountered in thelithographic fabrication of integrated circuits without departing fromthe spirit or scope of the present invention.

Other forms of interferometer described in copending commonly owned U.S.patent applications with Ser. No. 09/157,131 filed Sep. 18, 1998entitled “Interferometer Having A Dynamic Beam Steering Assembly” byHenry A. Hill and Peter de Groot; Ser. No. 09/305,828 filed May 5, 1999entitled “Interferometry System Having A Dynamic Beam Steering AssemblyFor Measuring Angle And Distance” by Henry A. Hill; Ser. No. 09/384,742filed Aug. 27, 1999 entitled “Polarization Preserving Optical Systems”by Henry A. Hill; Ser. No. 09/384,609 filed Aug. 27, 1999 entitled“Interferometer Having Reduced Ghost Beam Effects” by Peter de Groot;and Ser. No. 09/384,855 filed Aug. 27, 1999 entitled “InterferometersUtilizing Polarization Preserving Optical Systems” by Henry A. Hill maybe incorporated into the apparatus of the present invention, theforegoing applications being herein incorporated by reference, withoutdeparting from the spirit or scope of the present invention.

FIG. 2c depicts in schematic form, in accordance with the preferredapparatus and method of a variant of the first embodiment, electronicprocessor 127A. The variant of the first embodiment is from the firstcategory of embodiments of the several different categories andcomprises beam 109, source of beam 109, interferometer 169, detectorsystem 189, and digital computer 129 of the first embodiment shown inFIG. 2a and electronic processor 127A shown in FIG. 2c.

Electronic processor 127A comprises certain elements that perform likefunctions as like numbered elements of electronic processor 127 of thefirst embodiment. In the operation of electronic processor 127A, asshown in FIG. 3c, signal (s₂−s_(2,ψ,M)) generated by electronicprocessor 152 is transmitted to electronic processor 156 and electronicprocessor 151B. Electronic processor 151B generates a Fourier transformF(s₂−s_(2,ψ,M)) of (s₂−s_(2,ψ,M)) and F(s₂−s_(2,ψ,M)) is transmitted toelectronic processor 153 of 127A.

Non-zero complex spectral coefficients in F(s₂−s_(2,ψ,M)) representincomplete compensation for coherent cyclic errors in s₂. Incompletecompensation for coherent cyclic errors in s₂ can be the result forexample of changes of coherent cyclic errors with respect to time and/orthe result of statistical errors in measured quantities. Incompletecompensation will generally be present during an initialization phasefor establishing a multidimensional array of normalized, filtered, andinterpolated complex spectral coefficients by electronic processor 154A.Electronic processor 153 of 127A determines the complex spectralcoefficients of F(s₂−s_(2,ψ,M)) and the complex spectral coefficientsare transmitted to electronic processor 154A. Electronic processor 154Aprocesses the complex spectral coefficients representing an incompletecompensation of coherent cyclic errors in s₂ and updates themultidimensional array of normalized, filtered, and interpolated complexspectral coefficients. Electronic processor 154A otherwise processes themultidimensional array of normalized, filtered, and interpolated complexspectral coefficients the same as electronic processor 154 of processor127 of the first embodiment.

The remaining description of the variant of the first embodiment is thesame as corresponding portions of the description given for the firstembodiment.

FIG. 3a depicts in schematic form an apparatus and method in accordancewith the second embodiment of the present invention. The secondembodiment is from the second category of embodiments. Theinterferometer depicted in FIG. 3a is a polarizing, heterodyne, singlepass interferometer. Although the described embodiment is a heterodynesystem, the instant invention is readily adapted for use in a homodynesystem in which the reference and measurement beams have the samefrequencies.

Descriptions of a source of beam 209 and of beam 209 of the secondembodiment are the same as the corresponding portions of thedescriptions given for the source of beam 109 and of beam 109 of thefirst embodiment. Also descriptions of interferometer 269 and detectorsystem 289 of the second embodiment are the same as correspondingportions of the descriptions given for interferometer 169 and detectorsystem 189 of the first embodiment. Elements of the second embodimentcomprising the source of beam 209 and beam 209, interferometer 269, anddetector system 289 perform like functions as like numbered elements,decremented by 100, of the first embodiment comprising the source ofbeam 109 and beam 109, interferometer 169, and detector system 189.

Interferometer 269 introduces phase shift φ₃ between the first andsecond components of beam 241 so that beam 241 is a phase-shifted beam.The magnitude of phase shift φ₃ is related to round-trip physical lengthL₃ of measurement path 298 according to the formula

φ₃ =L ₃ p ₃ k ₃ n ₃  (28)

where p₃ is the number of passes through the respective reference andmeasurement legs, n₃ is the refractive index of a gas in measurementpath 298 corresponding to the optical path introducing the phase shiftφ₃ and to wavenumber k₃=2π/λ₃, and λ₃ is the wavelength of beam 207. Theinterferometer shown in FIG. 3a is for p₃=1 so as to illustrate in thesimplest manner the function of the apparatus of the second embodiment.To those skilled in the art, generalization to the case when p₃≠1 is astraight forward procedure. The value for L₃ corresponds to twice thedifference between the physical length of measurement path 298 and anassociated reference path.

In a next step as shown in FIG. 3a, phase-shifted beam 241 passesthrough polarizer 279, impinges upon photodetector 285, and generates anelectrical interference signal, heterodyne signal s₃, preferably byphotoelectric detection. Polarizer 279 is oriented so as to mixpolarization components of phase-shifted beam 241. Signal s₃ has theform

 s ₃ =A ₃(t)cos[α₃(t)]  (29)

where A₃(t) and α₃(t) are the amplitude and phase of s₃, respectively.

The signal s₃ is the real part, ŝ_(3,R), of a complex signal ŝ₃ where s₃comprises a causal, stable, i.e., absolutely summable, real sequence.Thus, the Fourier transform S_(3,R)(iω) of s₃ completely defines S₃(iω)[see Chapter 10 “Discrete Hilbert Transforms” in Discrete-Time SignalProcessing, (Prentice Hall, 1989) by A. V. Oppenheim and R. W. Schafer]where

S ₃(iω)=S _(3,R)(iω)+S _(3,I)(iω),  (30)

S_(3,I)(iω) is the imaginary component of S₃(iω), ω is an angularfrequency, and i is the imaginary number {square root over ((−1+L ))}.S_(3,I)(i,ω) is related to S_(3,R)(iω) by a frequency response function,H(iω), i.e.

S _(3,I)(iω)=H(iω)S _(3,R)(iω)  (31)

where the frequency response function H(iω) is given by the equation$\begin{matrix}{{H\left( {\quad \omega} \right)} = \left\{ \begin{matrix}{{- },} & {{0 < \omega},} \\{,} & {\omega < 0.}\end{matrix} \right.} & (32)\end{matrix}$

Imaginary component ŝ_(3,I) of ŝ₃ is obtained from the inverse Fouriertransform of S_(3,I)(iω) with

ŝ _(3,I) =A ₃(t)sin[α₃(t)].  (33)

The phase α₃ can be obtained from ŝ_(3,R) and ŝ_(3,I) according to theformula $\begin{matrix}{{\alpha_{3}(t)} = {{\arctan \left( \frac{s_{3,I}}{s_{3,R}} \right)}.}} & (34)\end{matrix}$

Time-dependent argument α₃ is expressed in terms of other quantitiesaccording to the formula

α₃=2πf ₃ t+φ₃+ψ₃+ζ₃  (35)

where ψ₃ comprises the non-linear cyclic error terms and phase offset ζ₃comprises all contributions to phase α₃ that are not related orassociated with the optical path of the measurement path 298 orreference path and not related to non-linear cyclic errors. Heterodynesignal s₃ is transmitted to electronic processor 227 for analysis aselectronic signal 223 in either digital or analog format, preferably indigital format. Electronic signal 223 further comprises a Nyquistangular frequency ω_(3,Ny) determined by the sampling frequency of ananalog-to-digital converter used in the conversion of s₃ to a digitalformat.

The phase of driver 205 is transmitted by electrical signal S_(3,Ref),reference signal 221, in either digital or analog format, preferably indigital format, to electronic processor 227. A reference signal, analternative reference signal to reference signal 221, may also begenerated by an optical pick off means and detector (not shown infigures) by splitting off a portion of beam 209 with a non-polarizingbeam splitter, mixing the split-off portion of beam 209, and detectingthe mixed portion to produce an alternative heterodyne reference signal.

Non-linear cyclic error ψ₃ comprises terms generated by the samemechanisms as described for the first embodiment and accordingly will bereferred hereinafter as a coherent cyclic error function. A spectralrepresentation of the coherent cyclic error function ψ₃, in terms ofquantities such as φ₃, ω₃, and ω_(3,Ny), can be based on differentfamilies of orthogonal polynomials and functions. Two examples are aseries comprising Fourier sine and cosine functions and a seriescomprising Chebyshev polynomial functions. Without departing from thespirit and scope of the present invention, the Fourier sine and cosineseries spectral representation of ψ₃ will be used.

The properties of the spectral representation of ψ₃ are described interms of properties of signal s₃. Signal s₃ has a form the same as theform of s₂ given by Eq. (2) and accordingly is written as${s_{3} = {{a_{3,1,0,1,0}{\cos \left( {{\omega_{3}t} + \phi_{3} + \zeta_{3,1,0,1,0}} \right)}} + {\sum\limits_{u,u^{\prime},p,p^{+}}{a_{3,u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega_{3}t} + \omega_{u^{\prime}}^{\prime} + {p\quad \phi_{3}} - {p^{+}\phi_{3}^{+}} + \zeta_{3,u,u^{\prime},p,p^{+}}} \right)}}} + {\sum\limits_{q}{\left( a_{3,1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{3,1,0,1,0,q,q}{\cos \left\lbrack {{q\left( {{\omega_{3}t} + \phi_{3}} \right)} + \zeta_{3,1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{3,1,0,1,0,q,{q - 2}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega_{3}t} + \phi_{3}} \right)} + \zeta_{3,1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{3,1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega_{3}t} + \phi_{3}} \right)} + \zeta_{3,1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}} + \ldots}}\quad;$

u=0,1;

u′=0,1, . . . except u′≠0 if u=1, p=1, p⁺=0; ω′_(3,0)=0;

p,p⁺=0,1, . . . ; w_(3,1)/w_(3,2);

p⁺≠0 if p=1 and u=1;

w_(3,1), w_(3,2)=1,2, . . . ; w_(3,1)≠w_(3,2);

q=2,3, . . . ;

where

φ₃ ⁺ =L ₃ p ₃ k ₃ ⁺ n ₃,  (37)

k ₃ ⁺=2π[(1/λ₃)+(f ₃ /c)].  (38)

The terms in ψ₃ can be obtained from Eq. (36) by first writing s₃ in theform $\begin{matrix}{{s_{3} = {{A_{3}(t)}\begin{bmatrix}{{\cos \quad \psi_{3}{\cos \left( {{\omega_{3}t} + \phi_{3} + \zeta_{3}} \right)}} -} \\{\sin \quad \psi_{3}\sin \quad \left( {{\omega_{3}t} + \phi_{3} + \zeta_{3}} \right)}\end{bmatrix}}},} & (39)\end{matrix}$

using Eqs. (29) and (35), and then using coefficients of sin(ω₃t+φ₃+ζ₃)and cos(ω₃t+φ₃+ζ₃) terms generated in the writing of s₃ in the formgiven by Eq. (39) to compute ψ₃ as the −arctan of the ratio of thecoefficients of sin(ω₃t+φ₃+ζ₃) and cos(ω₃t+φ₃+ζ₃) terms. Leading termsin Eq. (36) written in the form of Eq. (39) using trigonometricidentities are $\begin{matrix}\begin{matrix}{s_{3} = \quad {{\cos \left( {{\omega_{3}t} + \phi_{3} + \zeta_{3}} \right)} \times}} \\{\quad {\begin{Bmatrix}{{a_{3,1,0,1,0}{\cos \left( {\zeta_{3,1,0,1,0} - \zeta_{3}} \right)}} +} \\{{\sum\limits_{u,u^{\prime},p,p^{+}}{a_{3,u,u^{\prime},p,p^{+}}{\cos \begin{bmatrix}{{\left( {u - 1} \right)\omega_{3}t} + {\omega_{3u^{\prime}}^{\prime}t} + {\left( {p - 1} \right)\phi_{3}} -} \\{{p^{+}\phi_{3}^{+}} + \left( {\zeta_{3,u,u^{\prime},p,p^{+}} - \zeta_{3}} \right)}\end{bmatrix}}}} +} \\{{\sum\limits_{q}{\left( a_{3,1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{3,1,0,1,0,q,q}{\cos \begin{bmatrix}{{\left( {q - 1} \right)\left( {{\omega_{3}t} + \phi_{3}} \right)} +} \\\left( {\zeta_{3,1,0,1,0,q,q} - \zeta_{3}} \right)\end{bmatrix}}} +} \\{{B_{3,1,0,1,0,q,{q - 2}}{\cos \begin{bmatrix}{{\left( {q - 3} \right)\left( {{\omega_{3}t} + \phi_{3}} \right)} +} \\\left( {\zeta_{3,1,0,1,0,q,{q - 2}} - \zeta_{3}} \right)\end{bmatrix}}} +} \\{\ldots +} \\{q_{R}B_{3,1,0,1,0,q,q_{R}}{\cos \left( {\zeta_{3,1,0,1,{0q},q_{R}} - \zeta_{3}} \right)}}\end{Bmatrix}}} +} \\\ldots\end{Bmatrix} -}} \\{\quad {{\sin \left( {{\omega_{3}t} + \phi_{3} + \zeta_{3}} \right)} \times}} \\{\quad {\begin{Bmatrix}{{a_{3,1,0,1,0}{\cos \left( {\zeta_{3,1,0,1,0} - \zeta_{3}} \right)}} +} \\{{\sum\limits_{u,u^{\prime},p,p^{+}}{a_{3,u,u^{\prime},p,p^{+}}{\sin \begin{bmatrix}{{\left( {u - 1} \right)\omega_{3}t} + {\omega_{3u^{\prime}}^{\prime}t} + {\left( {p - 1} \right)\phi_{3}} -} \\{{p^{+}\phi_{3}^{+}} + \left( {\zeta_{3,u,u^{\prime},p,p^{+}} - \zeta_{3}} \right)}\end{bmatrix}}}} +} \\{{\sum\limits_{q}{\left( a_{3,1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{3,1,0,1,0,q,q}{\sin \begin{bmatrix}{{\left( {q - 1} \right)\left( {{\omega_{3}t} + \phi_{3}} \right)} +} \\\left( {\zeta_{3,1,0,1,0,q,q} - \zeta_{3}} \right)\end{bmatrix}}} +} \\{{B_{3,1,0,1,0,q,{q - 2}}{\sin \begin{bmatrix}{{\left( {q - 3} \right)\left( {{\omega_{3}t} + \phi_{3}} \right)} +} \\\left( {\zeta_{3,1,0,1,0,q,{q - 2}} - \zeta_{3}} \right)\end{bmatrix}}} +} \\{\ldots +} \\{q_{R}B_{3,1,0,1,0,q,q_{R}}{\sin \left( {\zeta_{3,1,0,1,{0q},q_{R}} - \zeta_{3}} \right)}}\end{Bmatrix}}} +} \\\ldots\end{Bmatrix};}}\end{matrix} & (40)\end{matrix}$

u=0,1;

u′=0,1, . . . except u′≠0 if u=1, p=1, p⁺=0; ω′_(3,0)=0

p,p⁺=0,1, . . . ; w_(3,1)/w_(3,2);

p⁺≠0 if p=1 and u=1;

w_(3,1), w_(3,2)=1,2, . . . ; w_(3,1)≠w_(3,2);

q=2,3, . . . .

The terms in the spectral representation of ψ₃ are readily identified bynoting that ψ₃ is equal to −arctan of the ratio of the coefficients ofsin(ω₃t+ψ₃+ζ₃) and cos(ω₃t+ψ₃+ζ₃) terms in Eq. (40) and inspection ofthe properties of the coefficients of sin(ω₃t+ψ₃+ζ₃) and cos(ω₃t+ψ₃+ζ₃)terms in Eq. (40).

The coefficients of terms in the spectral representation of ψ₃ will ingeneral depend on the rate of change of a phase associated with the termas a result, for example, of properties of group delay experienced bythe heterodyne signal.

Referring to FIG. 3c, electronic processor 227 comprises phase detector250 to process heterodyne signal s₃ for phase α₃ by either digital oranalog signal processes, preferably digital processes, using time-basedphase detection such as a digital Hilbert transform phase detector [seesection 4.1.1 of “Phase-locked loops: theory, design, and applications”2nd ed. McGraw-Hill (New York) 1993, by R. E. Best], zero crossing phasedetectors, or the like. Electronic processor 227 further comprisesspectrum analyzers 251A and 251B that process reference signal s_(3,Ref)and heterodyne signal s₃, respectively, for ω₃ and {dot over (α)}₃,respectively. Angular frequency {dot over (α)}₃ is the angular frequencyof the dominant peak in the power spectrum of s₃. Spectrum analyzers251A and 251B are preferably based on a sliding window Fourier transformalgorithm. Phase α₃ and angular frequency ω₃ are transmitted toelectronic processor 252 where {tilde over (φ)}₃=α₃−ω₃t and the Fouriertransform of α₃, F(α₃), are computed.

In a next step in electronic processor 227, Fourier transform F(α₃) andangular frequencies {dot over (α)}₃, ω₃, and ω_(3,Ny) are transmitted toelectronic processor 253 where the complex spectral coefficients ofcoherent cyclic error terms in ψ₃ are extracted at angular frequencies{tilde over (ω)}_(3,v) and aliases {tilde over (ω)}_(3,v,A), of {tildeover (ω)}_(3,v). An amplitude of a cyclic error term in α₃ correspondsto the amplitude of a corresponding peak in an associated power spectrumand the phase of the cyclic error term in α₃ corresponds to the arctanof the ratio of the imaginary and real components of F(α₃) at theangular frequency of the corresponding peak in the associated powerspectrum. Angular frequencies {tilde over (ω)}_(3,v), where v is anindex parameter comprising u, u′, p, p⁺, and q, correspond to the set ofangular frequencies equal to derivatives, with respect to time, of thearguments of the sinusoidal factors of terms in the spectralrepresentation of ψ₃. Aliases {tilde over (ω)}_(3,v,A) are given by theformula $\begin{matrix}{{{\overset{\sim}{\omega}}_{3,v,A} = {{\left( {- 1} \right)^{r}{\overset{\sim}{\omega}}_{3,v}} - {\left\lbrack {{\left( {- 1} \right)^{r}\left( {r + \frac{1}{2}} \right)} - \left( \frac{1}{2} \right)} \right\rbrack \omega_{3,{Ny}}}}},\quad {r = 1},2,\ldots} & (41)\end{matrix}$

with

 rω _(3,Ny)<{tilde over (ω)}_(3,v)<(r+1)ω_(3,Ny).  (42)

In practice, the amplitudes and associated phases of cyclic error termsin ψ₃ need be extracted only for a small subset of the set of possible{tilde over (ω)}_(3,v) and {tilde over (ω)}_(3,v,A). The selection ofthe subset of the set of possible {tilde over (ω)}_(3,v) and {tilde over(ω)}_(3,v,A) may be guided by properties of certain coefficients ofsin(ω₃t+ψ₃+ζ₃) and cos(ω₃t+φ₃+ζ₃) terms in Eq. (40). However, as part ofan initialization procedure, the selection of the subset of the set ofpossible {tilde over (ω)}_(3,v) and {tilde over (ω)}_(3,v,A) is based ona power spectrum analysis of α₃ and chi-square tests of peaks in thepower spectrum. The chi-square tests identify statistically significantpeaks in the power spectrum. Representation of the angular frequencies{tilde over (ω)}_(3,v) and {tilde over (ω)}_(3,v,A) of the subset of{tilde over (ω)}_(3,v) and {tilde over (ω)}_(3,v,A) associated with thestatistically significant peaks in terms of ω₃, ω_(3,Ny), {dot over(φ)}₃, {dot over (φ)}₃ ⁺, u, u′, p, p⁺, q, w_(3,1)/w_(3,2), and r isdetermined by observing properties of the respective {tilde over(ω)}_(3,v) and ω_(3,Ny) as {dot over (φ)}₃ is varied. Note that {dotover (φ)}₃ ⁺={dot over (φ)}₃ to a relative precision of the order of orless than 10⁻⁶.

The initialization procedure is performed by electronic processor 253.As part of an operating procedure of the second embodiment, powerspectrum analyses of α₃ and chi-square tests of peaks in the powerspectra are monitored for possible changes that may need be made to thesubset of the set of possible {tilde over (ω)}_(3,v) and {tilde over(ω)}_(3,v,A) during operation of the apparatus and method of the secondembodiment. The power spectrum analyses of α₃ and associated chi-squaretests executed as part of the monitoring procedure are also performed asa background task by electronic processor 253.

The extracted complex spectral coefficients in F(α₃) corresponding tocyclic error terms in ψ₃ are then sent to electronic processor 254 wherethe extracted spectral coefficients are normalized, filtered withrespect to time, and interpolations made as required and amultidimensional array of normalized, filtered, and interpolated complexspectral coefficients maintained. The step of normalization is for thepurpose of compensating for effects of non-zero values of second andhigher order derivatives of φ₂ with respect to time that exist at thetime of a determination of set of complex spectral coefficients. Thedimensionality of the multidimensional array is determined in part bythe magnitude of the filtered complex spectral coefficients, therequired precision of an end use application with respect to correctionfor coherent cyclic errors, and the dependence of the filtered complexspectral coefficients on {dot over (φ)}₂ and other system properties.

Electronic procession 255, in the next step in electronic processor 227as shown in FIG. 3b , computes the coherent cyclic error correctionψ_(3,M) using information listed in the multidimensional array ofnormalized, filtered, and interpolated spectral coefficients generatedby electronic processor 254. Electronic processor 256 (see FIG. 3b)computes φ₃ wherein the coherent cyclic errors have been compensated bysubtracting ψ_(3,M) from {tilde over (φ)}₃. Phase φ₃ is transmitted byelectronic processor 227 as signal 228 to computer 229 for use indownstream applications.

Each of the electronic processors comprising electronic processor 227preferably performs respective functions as digital processes. TheFourier transform of α₃ comprises Fourier transforms of (φ₃+ω₃t) andterms having factors such as cos[(u−1)ω₃t+ω′_(3u′)+(p−1)φ₃−p⁺φ₃ ⁺] andsin[(u−1)ω₃t+ω′_(3u′)+(p−1)φ₃−p⁺φ₃ ⁺] as evident from inspection of Eq.(40). The Fourier transform of (φ₃+ω₃t) over time interval T−τ/2 toT+τ/2, with φ₃ represented as a Taylor's series about t=T, may beexpressed as $\begin{matrix}{{F\left( {\phi_{3} + {\omega_{3}t}} \right)} = {{\frac{\tau}{\sqrt{2\pi}}{^{{\omega}\quad T}\left\lbrack {{\phi_{3}(T)} + {\omega_{3}T}} \right\rbrack}{j_{0}\left( \frac{\omega\tau}{2} \right)}} + \begin{matrix}{\frac{\tau}{\sqrt{2\pi}}^{{\omega}\quad T}{\begin{Bmatrix}{{{g_{1}\left( \frac{\omega\tau}{2} \right)}\left( \frac{\tau}{2} \right){{\overset{.}{\phi}}_{3}(T)}} + {{g_{2}\left( \frac{\omega\tau}{2} \right)}\left( \frac{1}{2!} \right)\left( \frac{\tau}{2} \right)^{2}{{\overset{¨}{\phi}}_{3}(T)}} +} \\{{{g_{3}\left( \frac{\omega\tau}{2} \right)}\left( \frac{1}{3!} \right)\left( \frac{\tau}{2} \right)^{3}{{\overset{...}{\phi}}_{3}(T)}} + {{g_{4}\left( \frac{\omega\tau}{2} \right)}\left( \frac{1}{4!} \right)\left( \frac{\tau}{2} \right)^{4}{{\overset{IV}{\phi}}_{3}(T)}} +} \\{{{g_{5}\left( \frac{\omega\tau}{2} \right)}\left( \frac{1}{5!} \right)\left( \frac{\tau}{2} \right)^{5}{{\overset{V}{\phi}}_{3}(T)}} + \ldots}\end{Bmatrix}.}}\end{matrix}}} & (43)\end{matrix}$

It is evident from inspection of Eq. (40) that, in the secondembodiment, there will be present zero frequency shift coherent cyclicerrors arising from terms having coefficients with factorsB_(3,1,0,1,0,q,q) _(R) , q=3, 5, . . . . The zero frequency shiftcoherent cyclic errors arise from the same type terms as the zerofrequency shift coherent cyclic errors present in the first embodiment.

A second embodiment procedure used for detecting and isolating theprimary zero frequency shift coherent cyclic errors is based on the sametechnique described for detecting and isolating the primary zerofrequency shift coherent cyclic errors for the first embodiment. Asecond beam 209T (not shown in FIG. 3a ) is introduced and used togenerate a multiplet in a power spectrum of phase α₃ centered at afrequency equal to one half the frequency spacing of contiguous membersof the multplet. The description of beam 209T is the same as thecorresponding portion of the description given for beam 109T of thefirst embodiment. Measured properties with respect to amplitudes andphases of members of the multiplet are processed by electronic processor227 to determine for the second embodiment the primary zero frequencyshift coherent cyclic errors following the same procedure as describedfor corresponding portions of the description given for the firstembodiment.

The remaining description of the second embodiment is the same ascorresponding portions of the description given for the firstembodiment.

FIG. 3c depicts in schematic form, in accordance with the preferredapparatus and method of a variant of the second embodiment, electronicprocessor 227A. The variant of the second embodiment is from the secondcategory of embodiments and comprises beam 209, source of beam 209,interferometer 269, detector system 289, and digital computer 229 of thesecond embodiment shown in FIG. 3a and electronic processor 227A shownin FIG. 3c.

Electronic processor 227A comprises certain elements that perform likefunctions as like numbered elements of electronic processor 227 of thesecond embodiment. In the operation of electronic processor 227A, asshown in FIG. 3c, phase α₃, computed coherent cyclic error ψ_(3,M), andangular frequency ω₃ are transmitted to electronic processor 257 whereφ₃, (α₃−ψ_(3,M)), and the Fourier transform of (α₃−ψ_(3,M)),F(α₃−ψ_(3,M)), are generated. Fourier transform F(α₃−ψ_(3,M)) istransmitted to electronic processor 253 of 227A along with angularfrequencies {dot over (α)}₃ from electronic processor 251B and ω_(3,Ny).Phase φ₃ is transmitted by electronic processor 227A as electronicsignal 228 to digital computer 229.

Non-zero spectral coefficients in F(α₃−ψ_(3,M)) represent incompletecompensation for coherent cyclic errors in φ₃. Incomplete compensationfor coherent cyclic errors in φ₃ can be a result for example of changesof coherent cyclic errors in time and/or the result of statisticalerrors in measured quantities. Incomplete compensation will generally bepresent during an initialization procedure for establishing amultidimensional array of normalized, filtered, and interpolatedspectral coefficients by electronic processor 254A. Electronic processor253 of 227A determines spectral coefficients from F(α₃−ψ_(3,M)) and thespectral coefficients are transmitted to electronic processor 254A.Electronic processor 254A processes the spectral coefficientsrepresenting incomplete compensation of coherent cyclic errors in φ₃ andupdates the multidimensional array of filtered coherent cyclic errorcoefficients. Electronic processor 254A further processes themultidimensional array of filtered coherent cyclic error coefficientsthe same as electronic processor 254 of processor 227 of the secondembodiment with respect to required corrections for non-zero second andhigher order derivatives of φ₃ and the identification and omission ofspectral coefficients corresponding to superimposed values, with respectto frequency, of respective coherent cyclic errors.

The remaining description of the variant of the second embodiment is thesame as corresponding portions of the description given for the secondembodiment.

FIGS. 4a, 4 b, and 4 c depict in schematic form in accordance with thepreferred apparatus and method of the third embodiment of the presentinvention. The third embodiment is from the third category of theseveral different categories of embodiments. Certain embodiments of thethird category of embodiments comprise apparatus and methods formeasuring and correcting for cyclic errors in optical dispersion relatedsignals such as used to measure and correct for effects of a gas in ameasuring path of a distance measuring interferometer. Certain otherembodiments of the third category of embodiments comprise apparatus andmethods for measuring and correcting for cyclic errors in both adispersion related signal and a signal used for determination of changesin optical path length of a measurement path in a distance measuringinterferometer. The effects of cyclic errors in corrections for effectsof a gas in a measuring path are greater than the effects of cyclicerrors in the signal used for determination of changes in optical pathlength by one and a half or more orders of magnitude.

The third embodiment comprises apparatus and method for measuring andmonitoring the dispersion of a gas in a measurement path of a distancemeasuring interferometer and correcting for effects of the gas in themeasurement path of the distance measuring interferometer. The thirdembodiment further comprises apparatus and method for measuring andcorrecting for effects of cyclic errors in both an optical dispersionrelated signal such as used to measure the dispersion of a gas and asignal used for determination of changes in an optical path length of ameasurement path in the distance measuring interferometer. Therefractive index of the gas and/or the physical length of themeasurement path may be changing. In addition, the ratio of wavelengthsof light beams generated by adopted light sources is matched with acertain relative precision to a known ratio value comprised of non-zeroquantities. The non-zero quantities may comprise one or more low ordernon-zero integers.

Referring to FIG. 4a and in accordance with the preferred apparatus andmethod of the third embodiment, the description of light beam 309 andthe source of light beam 309 is the same as corresponding portions ofthe descriptions given for light beam 109 and the source of light beam109 of the first embodiment. The wavelength of source 301 is λ₄. In anext step, a light beam 308 emitted from a source 302 passes through amodulator 304 becoming light beam 310. Modulator 304 is excited by anelectronic driver 306 similar to the excitation of modulator 303 byelectronic driver 305, respectively. Source 302, similar to source 301,is preferably a laser or like source of polarized, coherent radiation,but preferably at a different wavelength, λ₅.

The ratio of the wavelengths λ₄/λ₅ has a known approximate ratio valuel₄/l₅, i.e.

(λ₄/λ₅)≅(l ₄ /l ₅),  (44)

where l₄ and l₅ comprise non-zero quantities. The non-zero quantitiesmay comprise one or more low order non-zero integer values. Componentsof beams 309 and 310 with oscillation frequencies shifted by amounts f₄and f₅, respectively, with respect to non-frequency shifted componentsof beams 309 and 310, respectively, are polarized orthogonally to theplane of 4 a. Oscillation frequencies f₄ and f₅ are determined byelectronic drivers 305 and 306, respectively. In addition, thedirections of the frequency shifts of the frequency shifted componentsof beams 309 and 310 are the same.

It will be appreciated by those skilled in the art that beams 307 and308 may be provided alternatively be a single laser source emitting morethan one wavelength, by a single laser source combined with opticalfrequency doubling means to achieve frequency doubling, a laser sourcewith a non-linear element internal to the laser cavity, etc., two lasersources of differing wavelengths combined with sum-frequency generationor difference-frequency generation, or any equivalent sourceconfiguration capable of generating light beams of two or morewavelengths. It will also be appreciated by those skilled in the artthat one or both of the frequency shifts f₄ and f₅ may be the result ofZeeman splitting, birefringent elements internal to a laser cavity, orlike phenomena characteristic of the laser sources themselves. Thegeneration of beams by a single laser with two widely separatedwavelengths and for each beam, a pair of orthogonally polarizedcomponents, one component of each pair frequency shifted with respect tothe second component of the corresponding pair, is described in U.S.Pat. No. 5,732,095 entitled “Dual Harmonic-Wavelength Split-FrequencyLaser” and issued March 1998 to P. Zorabedian.

It will be further appreciated by those skilled in the art that bothpolarization components of beam 309 and/or of beam 310 may be frequencyshifted without departing from the scope and spirit of the invention, f₄remaining the difference in frequencies of the polarization componentsof beam 309 and f₅ remaining the difference in frequencies of thepolarization components of beam 310. Improved isolation of aninterferometer and a laser source is generally possible by frequencyshifting both polarization components of a beam, the degree of improvedisolation depending on the means used for generating the frequencyshifts.

In a next step, beam 309 is reflected by mirror 353A and a portionthereof reflected by dichroic non-polarizing beamsplitter 353B to becomea first component of beam 313, the λ₄ component. A portion of beam 310is transmitted by dichroic non-polarizing beamsplitter 353B to become asecond component of beam 313, the λ₅ component, wherein the propagationof the λ₅ component is preferably parallel and coextensive with thepropagation of the λ₄ component. In a further step, beam 313 propagatesto an interferometer 369 comprised of optical means for introducing aphase shift φ₄ between the non-frequency shifted and frequency shiftedcomponents of the λ₄ component of beam 313 and a phase shift φ₅ betweenthe non-frequency shifted and frequency shifted components of the λ₅component of beam 313. Magnitudes of phase shifts φ₄ and φ₅ are relatedto round-trip physical lengths L₄ and L₅ of measurement path 398according to the formulae

φ_(m) =L _(m) p _(m) k _(m) n _(m),  (45)

m=4 and 5,

where p_(m) is the number of passes through the respective reference andmeasurement legs for a multiple pass interferometer and n_(m) is therefractive index of a gas in measurement path 398 corresponding towavenumber k_(m)=(2π)/λ_(m).

As shown in FIG. 4a, interferometer 369 comprises a referenceretroreflector 391, object retroreflector 392 with a position controlledby translator 367, quarter-wave phase retardation plates 377 and 378,and a polarizing beam splitter 373. Quarter-wave phase retardationplates 377 and 378 and polarizing beam splitter 373 exhibit respectiveproperties at both λ₄ and λ₅. This configuration is known in the art asa polarized Michelson interferometer and is shown as a simpleillustration with p₄=p₅=1.

The number-of-passes parameters p₄ and p₅ may have values that are thesame or values that are different one with respect to other.

Eqs. (45) are valid for the case where the paths in an interferometerfor beams with one wavelength and the paths in the interferometer forbeams with the second wavelength are substantially coextensive, a casechosen to illustrate in the simplest manner the function of theinvention in the third embodiment. To those skilled in the art, thegeneralization to the case where the respective paths for beams with thetwo different wavelengths are not substantially coextensive is astraight-forward procedure.

After passing through interferometer 369, the portion of beam 313passing through the measurement path 398 becomes a phase-shifted beam333 and the portion of beam 313 passing through the reference pathcontaining retroreflector 391 becomes phase-shifted beam 334.Phase-shifted beams 333 and 334 are polarized orthogonal to the planeand in the plane of FIG. 4a, respectively. A conventional dichroic beamsplitter 361 separates those portions of beam 333 corresponding towavelengths λ₄ and λ₅ into beams 335 and 337, respectively, and thoseportions of beam 334 corresponding to wavelengths λ₄ and λ₅ into beams336 and 338, respectively. Beams 335 and 336 enter detector system 389and beams 337 and 338 enter detector system 390.

In detector system 389 as shown in FIG. 4a, beam 335 is first reflectedby mirror 363A and then reflected by polarizing beam splitter 363B toform a first component of beam 341. Beam 336 is transmitted bypolarizing beam splitter 363B to become a second component of beam 341.In detector system 390, beam 337 is first reflected by mirror 364A andthen reflected by polarizing beam splitter 364B to form a firstcomponent of beam 342. Beam 338 is transmitted by polarizing beamsplitter 364B to become a second component of beam 342. Beams 341 and342 pass through polarizers 379 and 380, respectively, impinge uponphotodetectors 385 and 386, respectively, and generate preferably byphotoelectric detection two electrical interference signals. The twoelectrical interference signals comprise two heterodyne signals s₄ ands₅, respectively. Polarizers 379 and 380 are preferably oriented so asto mix polarization components of beams 341 and 342, respectively. Theheterodyne signals s₄ and s₅ correspond to wavelengths λ₄ and λ₅,respectively.

Signals s₄ and s₅ have the forms${s_{m} = {{a_{m,1,0,1,0}{\cos \left( {{\omega_{m}t} + \phi_{m} + \zeta_{m,1,0,1,0}} \right)}} + {\sum\limits_{u,u^{\prime},p,p^{+}}{a_{m,u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega_{m}t} + {\omega_{{mu}^{\prime}}^{\prime}t} + {p\quad \phi_{m}} - {p^{+}\phi_{m}^{+}} + \zeta_{m,p,p^{+},u}} \right)}}} + {\begin{matrix}{{\sum\limits_{q}{\left( a_{m,1,0,1,0,} \right)^{q}\begin{Bmatrix}{{B_{m,1,0,1,q,q}{\cos \left\lbrack {{q\left( {{\omega_{m}t} + \phi_{m}} \right)} + \zeta_{m,1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{m,1,0,1,0,q,{q - 2}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega_{m}t} + \phi_{m}} \right)} + \zeta_{m,1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{m,1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega_{m}t} + \phi_{m}} \right)} + \zeta_{m,1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}} +}\end{matrix}\ldots}}}\quad;$

u=0,1;

u′=0,1, . . . except u′≠0 if u=1, p=1, p⁺=0; ω′_(4,0)=0;

p,p⁺=0, 1, . . . ; w_(m,1)/w_(m,2);

p⁺≠0 if p=1 and u=1;

w_(m,1), w_(m,2)=1,2, . . . ; w_(m,1)≠w_(m,2);

q=2,3, . . . ;

m=4,5;

where

 φ_(m) ⁺ =L _(m) p _(m) k _(m) ⁺ n _(m),  (47)

k _(m) ⁺=2π[(1/λ_(m))+(f _(m) /c)].  (48)

Descriptions of s₄ and s₅ representations given by Eqs. (46) are thesame as corresponding portions of the description given of the s₂representation of the first embodiment by Eq. (2). Heterodyne signals s₄and s₅ are transmitted to electronic processor 327 for analysis aselectronic signals 323 and 324, respectively, in either digital oranalog format, preferably in digital format.

Referring now to FIG. 4b, phase φ₄ is determined by certain elements ofelectronic processor 327 wherein the certain elements compriseelectronic processors 351A, 351B, 352A, 353A, 354A, 355A, and 356A thatperform like functions as elements 151A, 151B, 152, 153, 154, 155, and156, respectively, of the first embodiment. Phase φ₅ is determined bycertain other elements of electronic processor 327 wherein the certainother elements comprise electronic processors 351C, 351D, 352B, 353B,354B, 355B, and 356B (see FIG. 4c) that perform like functions aselements 151A, 151B, 152, 153, 154, 155, and 156, respectively, of thefirst embodiment.

The phases of electronic drivers 305 and 306 are transmitted byelectrical signals, reference signals 321 and 322, respectively, ineither digital or analog format, preferably in digital format, toelectronic processor 327. Electronic processors 351A and 351C processreference signals s_(4,Ref) and s_(5,Ref), respectively, to determineangular frequencies ω₄=2πf₄ and ω₅=2πf₅, respectively, preferably by asliding window FFT frequency detection algorithm.

Reference signals, alternatives to reference signals 321 and 322, mayalso be generated by an optical pick off means and detectors (not shownin figures) by splitting off portions of beams 309 and 310 with beamsplitters, preferably non-polarizing beam splitters, mixing therespective split-off portions of beam 309 and beam 310, and detectingthe mixed portions to produce alternative heterodyne reference signals.

Referring again to FIG. 4c, phases φ₄ and φ₅ are next multiplied byl₄/p₄ and l₅/p₅, respectively, in electronic processors 3275A and 3275B,respectively, preferably by digital processing, to produce phases(l₄/p₄)φ₄ and (l₅/p₅)φ₅, respectively. Phases (l₄/p₄)φ₄ and (l₅/p₅)φ₅are next added together in electronic processor 3276 and subtracted onefrom the other in electronic processor 3277, preferably by digitalprocesses, to create the phases θ and Φ, respectively. Formally,$\begin{matrix}{{\vartheta = \left( {{\frac{l_{5}}{p_{5}}\phi_{5}} + {\frac{l_{4}}{p_{4}}\phi_{4}}} \right)},} & (49) \\{\Phi = {\left( {{\frac{l_{5}}{p_{5}}\phi_{5}} - {\frac{l_{4}}{p_{4}}\phi_{4}}} \right).}} & (50)\end{matrix}$

Phases θ and Φ can also be written, using the definitions given by Eqs.(45), (49), and (50) as $\begin{matrix}{{\vartheta = \begin{Bmatrix}{{\overset{\_}{L}\left\lbrack {{\chi \left( {n_{5} + n_{4}} \right)} + {K\left( {n_{5} - n_{4}} \right)}} \right\rbrack} +} \\{{\Delta \quad {L\left\lbrack {{\chi \left( {n_{5} - n_{4}} \right)} + {K\left( {n_{5} + n_{4}} \right)}} \right\rbrack}} + \xi_{\zeta}}\end{Bmatrix}},} & (51) \\{{\Phi = \begin{Bmatrix}{{\overset{\_}{L}\left\lbrack {{\chi \left( {n_{5} - n_{4}} \right)} + {K\left( {n_{5} + n_{4}} \right)}} \right\rbrack} +} \\{{\Delta \quad {L\left\lbrack {{\chi \left( {n_{5} + n_{4}} \right)} + {K\left( {n_{5} - n_{4}} \right)}} \right\rbrack}} + Z_{\zeta}}\end{Bmatrix}},} & (52)\end{matrix}$

where

χ=(l ₅ k ₅ +l ₄ k ₄)/2,  (53)

K=(l ₅ k ₅ −l ₄ k ₄)/2,  (54)

{overscore (L)}=(L ₅ +L ₄)/2,  (55)

ΔL=(L ₅ −L ₄)/2,  (56)

$\begin{matrix}{{\xi_{\zeta} = \left( {{\frac{l_{5}}{p_{5}}\zeta_{5,1,0,1,0}} + {\frac{l_{4}}{p_{4}}\zeta_{4,1,0,1,0}}} \right)},} & (57) \\{Z_{\zeta} = {\left( {{\frac{l_{5}}{p_{5}}\zeta_{5,1,0,1,0}} - {\frac{l_{4}}{p_{4}}\zeta_{4,1,0,1,0}}} \right).}} & (58)\end{matrix}$

The preferred nominal value for ΔL is zero and for coextensive beamcomponents at wavelengths λ₄ and λ₅, ΔL<<λ₄ or λ₅.

Dispersion (n₅−n₄) of the gas corrected for effects of cyclic errors indispersion related signals can be determined from θ and Φ using theformula $\begin{matrix}\begin{matrix}{\left( {n_{5} - n_{4}} \right) = \quad {{\frac{1}{\chi \quad {\overset{\_}{L}\left\lbrack {1 - \left( {K/\chi} \right)^{2}} \right\rbrack}}\left\lbrack {\Phi - {\vartheta \left( {K/\chi} \right)} - Q_{\zeta}} \right\rbrack} -}} \\{\quad {{\left( \frac{\Delta \quad L}{\overset{\_}{L}} \right)\left( \frac{1}{n_{5} + n_{4}} \right)},}}\end{matrix} & (59)\end{matrix}$

where

Q _(ζ) =Z _(ζ)−(K/χ)ξ_(ζ).  (60)

For those applications related to distance measuring interferometry,heterodyne phase φ₄ and phases θ and Φ may be used to determine distanceL₄ as a quantity independent of the effects of the refractive index ofthe gas in the measuring path of the distance measuring interferometerand corrected for the effects of cyclic errors in both the dispersionrelated signals and optical path length related signals using theformula $\begin{matrix}{L_{4} = {{\frac{1}{\left( {\chi - K} \right)}\begin{Bmatrix}{{\frac{l_{4}}{p_{4}}\left( {\phi_{4} - \zeta_{4}} \right)} -} \\{\left\lbrack {1 - \left( {\Delta \quad {L/\overset{\_}{L}}} \right)} \right\rbrack {\frac{\Gamma}{\left\lbrack {1 + \left( {K/\chi} \right)} \right\rbrack}\left\lbrack {\Phi - {\left( {K/\chi} \right)\vartheta} - Q_{\zeta}} \right\rbrack}}\end{Bmatrix}} + {\Gamma \frac{1}{\left( {n_{5} + n_{4}} \right)}\Delta \quad {L\left\lbrack {1 - \left( {\Delta \quad {L/\overset{\_}{L}}} \right)} \right\rbrack}}}} & (61)\end{matrix}$

where Γ, the reciprocal dispersive power of the gas, is defined as$\begin{matrix}{\Gamma = {\frac{\left( {n_{4} - 1} \right)}{\left( {n_{5} - n_{4}} \right)}.}} & (62)\end{matrix}$

It is evident from the definition of K given by Eq. (54) that (K/χ)=0corresponds to wavelengths λ₄ and λ₅ being strictly harmonicallyrelated. For an application where |K/χ|>0 and the value of (K/χ) must beknown to a certain precision in the use of Eqs. (59) and/or (61) to meetan end use requirement, (K/χ) is measured by wavelength monitors. Thewavelength monitors may comprise interferometers with or without vacuumcells and/or frequency doubling of light beams by SHG. For anapplication where the value of χ must be known to another certainprecision in the use of Eqs. (59) and/or (61), χ is measured by awavelength monitor. In addition, when values for χ and (K/χ) are bothrequired, they may both be obtained from the same apparatus.

A value for the reciprocal dispersive power Γ can be obtained to acertain relative precision from known refractive properties of knownconstituents of a gas in the measuring path. For those applicationswhere the gas composition is not known to a requisite accuracy and/orthe refractive properties of the gas constituents is not known to acorresponding requisite accuracy, Γ can be measured by apparatus such asdescribed in copending commonly owned U.S. application Ser. No.09/232,515 filed Jan. 19, 1999 entitled “Apparatus And Methods ForMeasuring Intrinsic Optical Properties Of A Gas” by Henry A. Hill, theforegoing application being incorporated herein by reference.

The relative precision to which the dispersion (n₅−n₄) can bedetermined, if cyclic error effects are not compensated, is limited inpart by the effect of cyclic errors. The correction for cyclic erroreffects that has been made in obtaining the [Φ−θ(K/χ)−Q_(ζ)] factor inEqs. (59) and (61) is Q_(ψ) given by the formula

Q _(ψ) =Z _(ψ)−(K/χ)ξ₁₀₄   (63)

where $\begin{matrix}{{\xi_{\psi} = \left( {{\frac{l_{5}}{p_{5}}\psi_{5}} + {\frac{l_{4}}{p_{4}}\psi_{4}}} \right)},} & (64) \\{Z_{\psi} = {\left( {{\frac{l_{5}}{p_{5}}\psi_{5}} - {\frac{l_{4}}{p_{4}}\psi_{4}}} \right).}} & (65)\end{matrix}$

The correction Q_(ψ) enters as a term in a factor [{tilde over(Φ)}−{tilde over (θ)}(K/χ)−Q_(ψ)−Q_(ζ)] where {tilde over (Φ)} and{tilde over (θ)} are the corresponding values of Φ and θ, respectively,uncompensated for effects of cyclic errors. Thus the magnitude of thecyclic error effects in determined values of the dispersion (n₅−n₄),according to Eqs. (59) and (63), is of the order of $\begin{matrix}{{{\left\lbrack {\left( \frac{l_{m}}{p_{m}} \right){\psi_{m}}} \right\rbrack/\overset{\_}{L}}{\chi \left( {n_{5} - n_{4}} \right)}},{m = {4\quad {and}\quad 5.}}} & (66)\end{matrix}$

Consider for example, an application where λ₄=0.633 μm, λ₄=2λ₅, p₄=p₅=1,{overscore (L)}=0.5 m, and the gas is comprised of air at 25° C. and apressure of one atmosphere. For the example conditions, the magnitude ofthe contribution of ψ₄ to the relative precision as expressed by Eq.(66) is

≈0.019|ψ₄|,  (67)

ψ₄ being expressed in radians and |ψ₄| indicating the absolute value ofψ₄. Continuing with the example, for a specific cyclic error of |ψ₄|=0.1radians, a cyclic error in phase corresponding in the example to acyclic error in a distance measurement of 5 nm, the specific cyclicerror limits the relative precision to which the dispersion (n₅−n₄) canbe measured to ≈0.2%. If a source for the λ₄ beam is a NbYAG laser withλ₄=1.06 μm, the corresponding limits on the relative precision to whichthe dispersion (n₅−n₄) can be measured is ≈0.6%.

The limitations of effects of cyclic errors on the relative precision towhich the dispersion (n₅−n₄) can be determined, if cyclic error effectsare not compensated, propagate directly to limitations of effects ofcyclic errors on relative precision to which refractivity effects of gasin a measurement path of a distance measuring interferometer can bedetermined using dispersion interferometry. From inspection of Eq. (61),it is evident that the magnitude of the cyclic error contribution ofψ_(m) entering through Q_(ψ) is ≅Γ|ψ_(m)| relative to the magnitude ofthe cyclic error contribution |ψ₄| entering through the φ₄=({tilde over(φ)}₄−ψ₄−ζ₄) term. For the two cases of λ₄=0.633 μm with λ₄=2λ₅ andλ₄=1.06 μm also with λ₄=2λ₅, the values for Γ are 22 and 75,respectively. Thus the effects of cyclic error contributions to thecorrection term in Eq. (61) for the refractivity of a gas in a measuringpath must be reduced by approximately one and a half or more orders ofmagnitude if the effects of the cyclic error contributions resultingfrom the correction term are to be of the order of or less than theeffects of the cyclic error contributions resulting directly from theφ₄=({tilde over (φ)}₄−ψ₄−ζ₄) term.

The remaining description of the third embodiment is the same ascorresponding portions of the description given for the firstembodiment.

The distance L₅ can also be determined in a first variant of the thirdembodiment using φ₅ instead of φ₄. The corresponding equation for thedetermination of L₅ is $\begin{matrix}{L_{5} = {{\frac{1}{\left( {\chi + K} \right)}\begin{Bmatrix}{{\frac{l_{5}}{p_{5}}\left( {\phi_{5} - \zeta_{5}} \right)} -} \\{\left\lbrack {1 - \left( {\Delta \quad {L/\overset{\_}{L}}} \right)} \right\rbrack {\frac{\left( {\Gamma + 1} \right)}{\left\lbrack {1 - \left( {K/\chi} \right)} \right\rbrack}\left\lbrack {\Phi - {\left( {K/\chi} \right)\vartheta} - Q_{\zeta}} \right\rbrack}}\end{Bmatrix}} + {\frac{\left( {\Gamma + 1} \right)}{\left( {n_{5} + n_{4}} \right)}\Delta \quad {{L\left\lbrack {1 - \left( {\Delta \quad {L/\overset{\_}{L}}} \right)} \right\rbrack}.}}}} & (68)\end{matrix}$

The remaining description of the first variant of the third embodimentis the same as corresponding portions of the description given for thethird embodiment.

It will be evident to those skilled in the art that the quantities L₄and L₅ of the third embodiment and of the first variant of the thirdembodiment, respectively, can both be determined and used to obtainreduced statistical error in a determination of a change in a physicalpath length without departing from either the scope or spirit of thepresent invention.

It will also be evident to those skilled in the art that L₅ could be thedistance determined by the third embodiment instead of L₄ withoutdeparting from either the scope or spirit of the present invention.

It will be further evident to those skilled in the art that for thoseend use applications where K/χ must be known to a certain precisionand/or χ must be known to another certain precision, effects of cyclicerrors in measured wavelength and wavelength ratio values obtained bywavelength measuring and monitoring apparatus can be measured andcompensated by application of the apparatus and method of the first andsecond embodiments and variants thereof and wavelength monitors such assubsequently described herein in a sixth embodiment and variants thereofof the present invention without departing from the scope and spirit ofthe present invention.

For those end use applications where Γ is measured for the gas in themeasuring path 398, it may further be necessary to measure andcompensate for effects of cyclic errors. The subsequently describedherein in the sixth embodiment and variants thereof may be used inobtaining measured values of Γ compensated for effects of cyclic errors.

A second variant of the third embodiment is described. In the thirdembodiment, the effects of cyclic errors are compensated by usingcorresponding methods and apparatus of the first embodiment. In thesecond variant of the third embodiment, the effects of cyclic errors arecompensated by using the corresponding methods and apparatus of thevariant of the first embodiment. The remaining description of the secondvariant of the third embodiment is the same as corresponding portions ofdescriptions given for the first and third embodiments and variantsthereof.

The fourth embodiment of the present invention is described, inaccordance with the preferred apparatus and method of the fourthembodiment, which comprises apparatus and method for measuring andmonitoring the dispersion of a gas in a measurement path and the changein the optical path length of the measurement path due to the gas. Thefourth embodiment further comprises apparatus and method forcompensating for effects of cyclic errors on a measured dispersion ofthe gas and on measured changes in the optical path length of themeasurement path due to the gas. The fourth embodiment is from thefourth category of the several different categories. The refractiveindex of the gas and/or the physical length of the measurement path maybe changing. In addition, the ratio of the wavelengths of light beamsgenerated by adopted light sources is matched with a certain relativeprecision to a known ratio value comprised of non-zero quantities. Thenon-zero quantities may comprise one or more low order non-zerointegers.

In the third embodiment and variants thereof, effects of cyclic errorsare compensated by using corresponding methods and apparatus of thefirst embodiment and variants thereof, i.e. by compensating for effectsof cyclic errors in a space of electrical interference signals. In thefourth embodiment, effects of cyclic errors are compensated by using thecorresponding methods and apparatus of the second embodiment andvariants thereof, i.e. by compensating for effects of cyclic errors in aspace of phases of electrical interference signals.

The effects of cyclic errors in corrections for effects of a gas in ameasuring path, corrections generated from optical dispersion relatedsignals, are greater than the effects of cyclic errors in the signalused for determination of changes in optical path length by one and ahalf or more orders of magnitude.

In accordance with the preferred apparatus and method of the fourthembodiment, the fourth embodiment comprises light beam 309 and thesource of light beam 309, interferometer 369, translator 367, anddetector systems 389 and 390 of the third embodiment shown in FIG. 4a.The fourth embodiment further comprises electronic processor 427 shownin a FIGS. 5a and 5 b.

Electronic signals of the fourth embodiment corresponding to electronicsignals 321, 322, 323, and 324 shown FIG. 4a are hereinafter referencedas electronic signals 421, 422, 423, and 424, respectively. Thefrequencies of electronic drivers 305 and 306 of the fourth embodimentare f₆ and f₇, respectively, with angular frequencies ω₆ and ω₇,respectively. The angular frequencies ω_(6,Ny) and ω_(7,Ny) are theangular Nyquist frequencies of detector systems 389 and 390,respectively, for the fourth embodiment. Wavelengths of sources 301 and302 for the fourth embodiment are λ₆ and λ₇, respectively, with a knownapproximate ratio l₆/l₇. Reference signals s_(6,Ref) and s_(7,Ref) aretransmitted as electronic signals 421 and 422, respectively. Heterodynesignals s₆ and s₇ of the fourth embodiment correspond to heterodynesignals s₄ and s₅, respectively, of the third embodiment. Heterodynesignals s₆ and s₇ are transmitted as electronic signals 423 and 424,respectively.

Referring to FIG. 5a, phase {tilde over (φ)}₆ is determined by certainelements of electronic processor 427 wherein the certain elementscomprise electronic processors 450A, 451A, and 452A that perform likefunctions as electronic processors 250, 251A, and 252, respectively, ofthe second embodiment. Phase {tilde over (φ)}₇ is determined by certainother elements of electronic processor 427 wherein the certain otherelements comprise electronic processors 450B, 451C, and 452B performlike functions as electronic processors 250, 251A, and 252,respectively, of the second embodiment. Elements 4275A, 4275B, 4276, and4277 of the fourth embodiment perform like functions as elements 3275A,3275B, 3276, and 3277 of the third embodiment to determine {tilde over(θ)} and {tilde over (Φ)}. The definitions of {tilde over (θ)} and{tilde over (Φ)} of the fourth embodiment are $\begin{matrix}{{\overset{\sim}{\vartheta} = \left( {{\frac{l_{7}}{p_{7}}{\overset{\sim}{\phi}}_{7}} + {\frac{l_{6}}{p_{6}}{\overset{\sim}{\phi}}_{6}}} \right)},} & (69) \\{\overset{\sim}{\Phi} = {\left( {{\frac{l_{7}}{p_{7}}{\overset{\sim}{\phi}}_{7}} - {\frac{l_{6}}{p_{6}}{\overset{\sim}{\phi}}_{6}}} \right).}} & (70)\end{matrix}$

where {tilde over (φ)}₆=α₆−ω₆t and {tilde over (φ)}₇=α₇−ω₇t. Thedescriptions of α₆ and α₇ are the same as corresponding portions of thedescription given for α₃ of the second embodiment.

Electronic processor 427 further comprises electronic processors 452C,453, 454, and 455 to determine Z_(ψ,M) of Q_(ψ,M) wherein

 Q _(ψ) =Z _(ψ)−(K/χ)ξ_(ψ),  (71) $\begin{matrix}{{\xi_{\psi} = \left( {{\frac{l_{7}}{p_{7}}\psi_{7}} + {\frac{l_{6}}{p_{6}}\psi_{6}}} \right)},} & (72) \\{Z_{\psi} = {\left( {{\frac{l_{7}}{p_{7}}\psi_{7}} - {\frac{l_{6}}{p_{6}}\psi_{6}}} \right).}} & (73)\end{matrix}$

Electronic processors 452C, 453, 454, and 455 of the fourth embodimentperform like functions as electronic processors 252, 253, 254, and 255of the second embodiment. Phases {tilde over (Φ)} and Z_(ψ,M) aretransmitted to electronic processor 456 where {tilde over (Φ)}−Z_(ψ,M)is generated. Electronic processor 456 of the fourth embodiment performslike functions as electronic processor 256, of the second embodiment.

The cyclic error compensated phase {tilde over (Φ)}−Z_(ψ,M) istransmitted to digital computer 429 as signal 428. Digital computer 429computers dispersion (n₇−n₆) according to the formula $\begin{matrix}{\left( {n_{7} - n_{6}} \right) = {{\frac{1}{\chi \quad {\overset{\_}{L}\left\lbrack {1 - \left( {K/\chi} \right)^{2}} \right\rbrack}}\left\lbrack {\left( {\overset{\sim}{\Phi} - Z_{\psi}} \right) - {\left( {\overset{\sim}{\vartheta} - \xi_{\psi}} \right)\left( {K/\chi} \right)} - Q_{\zeta}} \right\rbrack} - {\left( \frac{\Delta \quad L}{\overset{\_}{L}} \right){\left( \frac{1}{n_{7} + n_{6}} \right).}}}} & {(74)\quad}\end{matrix}$

The effects of cyclic errors in {tilde over (θ)}, ξ_(ψ), are included inEq. (74) for completeness. However, the effects of ξ_(ψ) are notcompensated in the fourth embodiment. Note that the effect of ξ_(ψ) inthe computation of the dispersion (n₇−n₆) is reduced by the factor (K/χ)relative to the effect of Z_(ψ) and therefore the effect of ξ_(ψ) can benegligible for end use applications where |K/χ|<<1.

For those applications related to distance measuring interferometry,heterodyne phase {tilde over (φ)}₆ and phases {tilde over (θ)} and{tilde over (Φ)} may be used to determine distance L₆ as a quantityindependent of the effects of the refractive index of the gas in themeasuring path of the distance measuring interferometer and correctedfor effects of cyclic errors in the dispersion related signals using theformula $\begin{matrix}{L_{6} = \quad {{\frac{1}{\left( {\chi - K} \right)}\begin{Bmatrix}{{\frac{l_{6}}{p_{6}}\left( {{\overset{\sim}{\phi}}_{6} - \zeta_{6}} \right)} -} \\{\left\lbrack {1 - \left( {\Delta \quad {L/\overset{\_}{L}}} \right)} \right\rbrack {\frac{\Gamma}{\left\lbrack {1 + \left( {K/\chi} \right)} \right\rbrack}\left\lbrack {\left( {\overset{\sim}{\Phi} - Z_{\psi}} \right) - {\left( {K/\chi} \right)\left( {\overset{\sim}{\vartheta} - \xi_{\psi}} \right)} - Q_{\zeta}} \right\rbrack}}\end{Bmatrix}} + {\Gamma \frac{1}{\left( {n_{7} + n_{6}} \right)}\Delta \quad {{L\left\lbrack {1 - \left( {\Delta \quad {L/\overset{\_}{L}}} \right)} \right\rbrack}.}}}} & (75)\end{matrix}$

The remaining description of the fourth embodiment is the same ascorresponding portions of the descriptions given for the second andthird embodiments of the present invention.

FIGS. 5c and 5 d depict in schematic form, in accordance with thepreferred apparatus and method of the fifth embodiment of the presentinvention, electronic processor 527A. The fifth embodiment is from thefourth category of embodiments and comprises beam 309, source of beam309, interferometer 369, detector system 389, and digital computer 329of the third embodiment shown in FIG. 4a and electronic processor 427Ashown in FIGS. 5c and 5 d.

The fifth embodiment comprises apparatus and method for measuring andmonitoring dispersion of a gas in a measurement path, the change in theoptical path length of the measurement path due to the gas, and a changein the optical path length of the measurement path due to a change inthe physical path length of the measurement path. The fifth embodimentfurther comprises apparatus and method for measuring and compensatingfor effects of cyclic errors in the measured dispersion of a gas in themeasurement path, in the change in the optical path length of themeasurement path due to the gas, and in the change in the optical pathlength of the measurement path due to a change in the physical pathlength of the measurement path.

In the fourth embodiment, effects of cyclic errors are compensated byusing corresponding methods and apparatus of the first embodiment andvariants thereof. In the fifth embodiment, the effects of cyclic errorsare also compensated by using the corresponding methods and apparatus ofthe first embodiment and variants thereof.

Electronic processor 427A comprises electronic processors 450A, 451A,451B, 452A, 453A, 454A, 455A, and 456A that perform like functions aselectronic processors 250, 251A, 251B, 252, 253, 254, 255, and 256 ofthe second embodiment to generated ψ_(6,M) and {tilde over(φ)}₆−ψ_(6, M). Electronic processor 427A further comprises electronicprocessors 450B, 451C, 451D, and 452B that perform the like functions as250, 251A, 251B, and 252 to generate {tilde over (φ)}₇. Electronicprocessors 4275A, 4275B, 4276, and 4277 of the fifth embodiment performlike functions as electronic processors 453B, 454B, 455B, and 456B thatperform like functions as electronic processors 4275A, 4275B, 4276, and4277 of the fourth embodiment to generate {tilde over (θ)} and {tildeover (Φ)}. Electronic processors 452C, 453B, 454B, 455B, and 456Bperform like functions as electronic processors 452C, 453, 454, 455, and456 of the fourth embodiment to generate Z_(6,M) and {tilde over(Φ)}−Z_(6,M),

The cyclic error corrected phase {tilde over (φ)}₆−ψ_(6,M), phases{tilde over (θ)} and {tilde over (Φ)}−Z_(ψ,M), and cyclic errorcorrection terms ψ_(6,M) and Z_(ψ,M) are transmitted to digital computer329 as signal 428A and used by digital computer 329 to determinedistance L₆ as a quantity independent of the effects of the refractiveindex of the gas in the measuring path of the distance measuringinterferometer and compensated for effects of cyclic errors in both thedispersion and distance measuring related signals using Eq. (75) whereinξ_(ψ,M) is computed by the formula $\begin{matrix}{\xi_{\psi,M} = {Z_{\psi,M} + {2\left( \frac{l_{6}}{p_{6}} \right){\psi_{6,M}.}}}} & (76)\end{matrix}$

The remaining description of the fifth embodiment is the same ascorresponding portions of the descriptions given for the second, third,and fourth embodiments and variants therein.

A sixth embodiment is described, in accordance with the preferredapparatus and method of the sixth embodiment, that comprises both anapparatus and method for measuring and correcting for cyclic errors inboth a dispersion measuring related signal and a refractivity measuringrelated signal or refractivity measuring related signals used todetermine intrinsic optical properties of a gas. The sixth embodimentfurther comprises both an apparatus and method for measuring andcorrecting for cyclic errors in a wavelength measuring and/or relatedsignal used to determine and/or monitor the wavelength of an opticalbeam. The sixth embodiment is from the fifth category of the severaldifferent categories.

The apparatus and method of the sixth embodiment comprises apparatus andmethod of the fourth embodiment to measure a dispersion of a gas and arefractivity of the gas to determine a corresponding reciprocaldispersal power of the gas that is corrected for effects of cyclicerrors. For the determination of the refractivity of the gas, a vacuumis provided for a measurement path at the respective wavelength. Thevacuum measurement path can also be used for measuring and monitoringthe wavelength compensated for effects of cyclic errors. The wavelengthof the second wavelength used to measure the dispersion can be measuredand monitored compensated for cyclic errors by providing a measurementpath at the second wavelength.

Reference is made to U.S. application Ser. No. 09/323,515 entitled“Apparatus And Methods For Measuring Intrinsic Optical Properties Of AGas,” ibid., for further description of apparatus and methods formeasuring and monitoring intrinsic optical properties of a gas andwavelengths of optical beams. The foregoing application has beenincorporated herein by reference.

The remaining description of the sixth embodiment is the same ascorresponding portions of the description given for the fourthembodiment and variants thereof.

A variant of the sixth embodiment is described, in accordance with thepreferred apparatus and method of the variant of the sixth embodiment,that comprises both an apparatus and method for measuring and correctingfor cyclic errors in both a dispersion measuring related signal and arefractivity measuring related signal or refractivity measuring relatedsignals used to determine intrinsic optical properties of a gas. Thevariant of the sixth embodiment further comprises both an apparatus andmethod for measuring and correcting for cyclic errors in a wavelengthmeasuring and/or related signal used to determine and/or monitor thewavelength of an optical beam. The variant of the sixth embodiment isfrom the fifth category of the several different categories.

The apparatus and method of the variant of the sixth embodimentcomprises apparatus and method of the fifth embodiment to measure adispersion of a gas and a refractivity of the gas to determine acorresponding reciprocal dispersal power of the gas that is correctedfor effects of cyclic errors. For the determination of the refractivityof the gas, a vacuum is provided for a measurement path at therespective wavelength. The vacuum measurement path can also be used formeasuring and monitoring the wavelength compensated for effects ofcyclic errors. The wavelength of the second wavelength used to measurethe dispersion can be measured and monitored compensated for cyclicerrors by providing a measurement path at the second wavelength.

The remaining description of the variant of the sixth embodiment is thesame as corresponding portions of the description given for the fifthand sixth embodiments.

The interferometry systems described quantify nonlinearities and use thequantified nonlinearities to correct distance measurements, dispersionmeasurements, and intrinsic optical property measurements for thepresence of such nonlinearities. As a result, such interferometrysystems provide highly accurate measurements. Such systems can beespecially useful in lithography applications used in fabricating largescale integrated circuits such as computer chips and the like.Lithography is the key technology driver for the semiconductormanufacturing industry. Overlay improvement is one of the five mostdifficult challenges down to and below 100 nm line widths (designrules), see for example the Semiconductor Industry Roadmap, p82 (1997).

Overlay depends directly on the performance, i.e. accuracy andprecision, of the distance measuring interferometers used to positionthe wafer and reticle (or mask) stages. Since a lithography tool mayproduce $50-100M/year of product, the economic value from improvedperformance distance measuring interferometers is substantial. Each 1%increase in yield of the lithography tool results in approximately$1M/year economic benefit to the integrated circuit manufacturer andsubstantial competitive advantage to the lithography tool vendor.

The function of a lithography tool is to direct spatially patternedradiation onto a photoresist-coated wafer. The process involvesdetermining which location of the wafer is to receive the radiation(alignment) and applying the radiation to the photoresist at thatlocation (exposure).

To properly position the wafer, the wafer includes alignment marks onthe wafer that can be measured by dedicated sensors. The measuredpositions of the alignment marks define the location of the wafer withinthe tool. This information, along with a specification of the desiredpatterning of the wafer surface, guides the alignment of the waferrelative to the spatially patterned radiation. Based on suchinformation, a translatable stage supporting the photoresist-coatedwafer moves the wafer such that the radiation will expose the correctlocation of the wafer.

During exposure, a radiation source illuminates a patterned reticle,which scatters the radiation to produce the spatially patternedradiation. The reticle is also referred to as a mask, and these termsare used interchangeably below. In the case of reduction lithography, areduction lens collects the scattered radiation and forms a reducedimage of the reticle pattern. Alternatively, in the case of proximityprinting, the scattered radiation propagates a small distance (typicallyon the order of microns) before contacting the wafer to produce a 1:1image of the reticle pattern. The radiation initiates photo-chemicalprocesses in the resist that convert the radiation pattern into a latentimage within the resist.

Interferometry systems are important components of the positioningmechanisms that control the position of the wafer and reticle, andregister the reticle image on the wafer. If such interferometry systemsinclude the phase measurement portion described above, the accuracy ofdistances measured by the systems increases as cyclic errorcontributions to the distance measurement are minimized.

In general, the lithography system, also referred to as an exposuresystem, typically includes an illumination system and a waferpositioning system. The illumination system includes a radiation sourcefor providing radiation such as ultraviolet, visible, x-ray, electron,or ion radiation, and a reticle or mask for imparting the pattern to theradiation, thereby generating the spatially patterned radiation. Inaddition, for the case of reduction lithography, the illumination systemcan include a lens assembly for imaging the spatially patternedradiation onto the wafer. The imaged radiation exposes resist coatedonto the wafer. The illumination system also includes a mask stage forsupporting the mask and a positioning system for adjusting the positionof the mask stage relative to the radiation directed through the mask.The wafer positioning system includes a wafer stage for supporting thewafer and a positioning system for adjusting the position of the waferstage relative to the imaged radiation. Fabrication of integratedcircuits can include multiple exposing steps. For a general reference onlithography, see, for example, J. R. Sheats and B. W. Smith, inMicrolithography: Science and Technology (Marcel Dekker, Inc., New York,1998), the contents of which is incorporated herein by reference.

Interferometry systems described above can be used to precisely measurethe positions of each of the wafer stage and mask stage relative toother components of the exposure system, such as the lens assembly,radiation source, or support structure. In such cases, theinterferometry system can be attached to a stationary structure and themeasurement object attached to a movable element such as one of the maskand wafer stages. Alternatively, the situation can be reversed, with theinterferometry system attached to a movable object and the measurementobject attached to a stationary object.

More generally, such interferometry systems can be used to measure theposition of any one component of the exposure system relative to anyother component of the exposure system, in which the interferometrysystem is attached to, or supported by, one of the components and themeasurement object is attached, or is supported by the other of thecomponents.

An example of a lithography scanner 1100 using an interferometry system1126 is shown in FIG. 6a. The interferometry system is used to preciselymeasure the position of a wafer (not shown) within an exposure system.Here, stage 1122 is used to position and support the wafer relative toan exposure station. Scanner 1100 includes a frame 1102, which carriesother support structures and various components carried on thosestructures. An exposure base 1104 has mounted on top of it a lenshousing 1106 atop of which is mounted a reticle or mask stage 1116,which is used to support a reticle or mask. A positioning system forpositioning the mask relative to the exposure station is indicatedschematically by element 1117. Positioning system 1117 can include,e.g., piezoelectric transducer elements and corresponding controlelectronics. Although, it is not included in this described embodiment,one or more of the interferometry systems described above can also beused to precisely measure the position of the mask stage as well asother moveable elements whose position must be accurately monitored inprocesses for fabricating lithographic structures (see supra Sheats andSmith Microlithography: Science and Technology).

Suspended below exposure base 1104 is a support base 1113 that carrieswafer stage 1122. Stage 1122 includes a plane mirror 1128 for reflectinga measurement beam 1154 directed to the stage by interferometry system1126. A positioning system for positioning stage 1122 relative tointerferometry system 1126 is indicated schematically by element 1119.Positioning system 1119 can include, e.g., piezoelectric transducerelements and corresponding control electronics. The measurement beamreflects back to the interferometry system, which is mounted on exposurebase 1104. The interferometry system can be any of the embodimentsdescribed previously.

During operation, a radiation beam 1110, e.g., an ultraviolet (UV) beamfrom a UV laser (not shown), passes through a beam shaping opticsassembly 1112 and travels downward after reflecting from mirror 1114.Thereafter, the radiation beam passes through a mask (not shown) carriedby mask stage 1116. The mask (not shown) is imaged onto a wafer (notshown) on wafer stage 1122 via a lens assembly 1108 carried in a lenshousing 1106. Base 1104 and the various components supported by it areisolated from environmental vibrations by a damping system depicted byspring 1120.

In other embodiments of the lithographic scanner, one or more of theinterferometry systems described previously can be used to measuredistance along multiple axes and angles associated for example with, butnot limited to, the wafer and reticle (or mask) stages. Also, ratherthan a UV laser beam, other beams can be used to expose the waferincluding, e.g., x-ray beams, electron beams, ion beams, and visibleoptical beams.

In some embodiments, the lithographic scanner can include what is knownin the art as a column reference. In such embodiments, theinterferometry system 1126 directs the reference beam (not shown) alongan external reference path that contacts a reference mirror (not shown)mounted on some structure that directs the radiation beam, e.g., lenshousing 1106. The reference mirror reflects the reference beam back tothe interferometry system. The interference signal produce byinterferometry system 1126 when combining measurement beam 1154reflected from stage 1122 and the reference beam reflected from areference mirror mounted on the lens housing 1106 indicates changes inthe position of the stage relative to the radiation beam. Furthermore,in other embodiments the interferometry system 1126 can be positioned tomeasure changes in the position of reticle (or mask) stage 1116 or othermovable components of the scanner system. Finally, the interferometrysystems can be used in a similar fashion with lithography systemsinvolving steppers, in addition to, or rather than, scanners.

As is well known in the art, lithography is a critical part ofmanufacturing methods for making semiconducting devices. For example,U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods.These steps are described below with reference to FIGS. 6b and 6 c. FIG.6b is a flow chart of the sequence of manufacturing a semiconductordevice such as a semiconductor chip (e.g. IC or LSI), a liquid crystalpanel or a CCD. Step 1151 is a design process for designing the circuitof a semiconductor device. Step 1152 is a process for manufacturing amask on the basis of the circuit pattern design. Step 1153 is a processfor manufacturing a wafer by using a material such as silicon.

Step 1154 is a wafer process which is called a pre-process wherein, byusing the so prepared mask and wafer, circuits are formed on the waferthrough lithography. To form circuits on the wafer that correspond withsufficient spatial resolution those patterns on the mask,interferometric positioning of the lithography tool relative the waferis necessary. The interferometry methods and systems described hereincan be especially useful to improve the effectiveness of the lithographyused in the wafer process.

Step 1155 is an assembling step, which is called a post-process whereinthe wafer processed by step 1154 is formed into semiconductor chips.This step includes assembling (dicing and bonding) and packaging (chipsealing). Step 1156 is an inspection step wherein operability check,durability check and so on of the semiconductor devices produced by step1155 are carried out. With these processes, semiconductor devices arefinished and they are shipped (step 1157).

FIG. 6c is a flow chart showing details of the wafer process. Step 1161is an oxidation process for oxidizing the surface of a wafer. Step 1162is a CVD process for forming an insulating film on the wafer surface.Step 1163 is an electrode forming process for forming electrodes on thewafer by vapor deposition. Step 1164 is an ion implanting process forimplanting ions to the wafer. Step 1165 is a resist process for applyinga resist (photosensitive material) to the wafer. Step 1166 is anexposure process for printing, by exposure (i.e., lithography), thecircuit pattern of the mask on the wafer through the exposure apparatusdescribed above. Once again, as described above, the use of theinterferometry systems and methods described herein improve the accuracyand resolution of such lithography steps.

Step 1167 is a developing process for developing the exposed wafer. Step1168 is an etching process for removing portions other than thedeveloped resist image. Step 1169 is a resist separation process forseparating the resist material remaining on the wafer after beingsubjected to the etching process. By repeating these processes, circuitpatterns are formed and superimposed on the wafer.

The interferometry systems described above can also be used in otherapplications in which the relative position of an object needs to bemeasured precisely. For example, in applications in which a write beamsuch as a laser, x-ray, ion, or electron beam, marks a pattern onto asubstrate as either the substrate or beam moves, the interferometrysystems can be used to measure the relative movement between thesubstrate and write beam.

As an example, a schematic of a beam writing system 1200 is shown inFIG. 7. A source 1210 generates a write beam 1212, and a beam focusingassembly 1214 directs the radiation beam to a substrate 1216 supportedby a movable stage 1218. To determine the relative position of thestage, an interferometry system 1220 directs a reference beam 1222 to amirror 1224 mounted on beam focusing assembly 1214 and a measurementbeam 1226 to a mirror 1228 mounted on stage 1218. Since the referencebeam contacts a mirror mounted on the beam focusing assembly, the beamwriting system is an example of a system that uses a column reference.Interferometry system 1220 can be any of the interferometry systemsdescribed previously. Changes in the position measured by theinterferometry system correspond to changes in the relative position ofwrite beam 1212 on substrate 1216. Interferometry system 1220 sends ameasurement signal 1232 to controller 1230 that is indicative of therelative position of write beam 1212 on substrate 1216. Controller 1230sends an output signal 1234 to a base 1236 that supports and positionsstage 1218. In addition, controller 1230 sends a signal 1238 to source1210 to vary the intensity of, or block, write beam 1212 so that thewrite beam contacts the substrate with an intensity sufficient to causephotophysical or photochemical change only at selected positions of thesubstrate.

Furthermore, in some embodiments, controller 1230 can cause beamfocusing assembly 1214 to scan the write beam over a region of thesubstrate, e.g., using signal 1244.

As a result, controller 1230 directs the other components of the systemto pattern the substrate. The patterning is typically based on anelectronic design pattern stored in the controller. In some applicationsthe write beam patterns a resist coated on the substrate and in otherapplications the write beam directly patterns, e.g., etches, thesubstrate.

An important application of such a system is the fabrication of masksand reticles used in the lithography methods described previously. Forexample, to fabricate a lithography mask an electron beam can be used topattern a chromium-coated glass substrate. In such cases where the writebeam is an electron beam, the beam writing system encloses the electronbeam path in a vacuum. Also, in cases where the write beam is, e.g., anelectron or ion beam, the beam focusing assembly includes electric fieldgenerators such as quadrapole lenses for focusing and directing thecharged particles onto the substrate under vacuum. In other cases wherethe write beam is a radiation beam, e.g., x-ray, UV, or visibleradiation, the beam focusing assembly includes corresponding optics andfor focusing and directing the radiation to the substrate.

Other aspects, advantages, and modifications are within the scope of thefollowing claims.

What is claimed is:
 1. An interferometry system comprising: aninterferometer which during operation directs two beams along separatepaths and then combines the beams to produce an overlapping pair of exitbeams, the separate paths defining an optical path length difference; adetector which responds to optical interference between the overlappingpair of exit beams and produces an interference signal s(t) indicativeof the optical path length difference, the signal s(t) including adominant term having a frequency equal to the sum of the frequencysplitting ω between the two beams, if any, and a Doppler shift {dot over(φ)} defined by the rate of change of the optical path lengthdifference, wherein properties of the interferometry system causes thesignal s(t) to further include additional terms each having a frequencynot equal to the sum of the frequency splitting ω and the Doppler shift{dot over (φ)}; and an analyzer coupled to the detector which duringoperation: i) quantifies at least one of the additional terms based onvalues of s(t) for which the value of the Doppler shift causes thedominant term and the at least one additional term to be separatedspectrally; and ii) uses the quantified at least one additional term toestimate a change in the optical path length difference corresponding toanother value of s(t) for which the value of the Doppler shift does notcauses the dominant term and the at least one additional term to overlapspectrally.
 2. The system of claim 1, wherein the detector comprises aphotodetector, an amplifier, and an analog-to-digital converter.
 3. Thesystem of claim 1, wherein the frequency splitting between the two beamsis nonzero.
 4. The system of claim 1, wherein the at least one of theadditional terms comprises a plurality of the additional terms.
 5. Theinterferometry system of claim 1, wherein to quantify the at least oneadditional term, the analyzer calculates the Doppler shift {dot over(φ)} for the values of s(t)based on the expressions(t)∝cos(ωt+φ+ζ_(1,0,1,0))+NL, where NL is an initial quantification ofthe additional terms, and where φ=Lkn, L is the physical path lengthdifference, k is a wavenumber, n is a refractive index, ω is the angularfrequency splitting between the two beams, t is time, and ζ_(1,0,1,0) isa phase-offset.
 6. The interferometry system of claim 5, wherein theinitial quantification is NL=0.
 7. The interferometry system of claim 1,wherein the analyzer quantifies the at least one additional term byestimating corresponding coefficients of a representation of s(t) thataccounts for the additional terms.
 8. The interferometry system of claim7, wherein the representation of s(t) can be expressed as:${s(t)} = {{a_{1,0,1,0}{\cos \left( {{\omega \quad t} + \phi + \zeta_{1,0,1,0}} \right)}} + {\sum\limits_{u,u^{\prime},p,p^{+}}{a_{u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega \quad t} + {\omega_{u^{\prime}}^{\prime}t} + {p\quad \phi} - {p^{+}\phi^{+}} + \zeta_{u,u^{\prime},p,p^{+}}} \right)}}} + {\sum\limits_{q}{\left( a_{1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{1,0,1,0,q,q}{\cos \left\lbrack {{q\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{1,0,1,0,q,{q - 2}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}}}$

+ . . . , u=0 or 1; u′=0,1, . . . ; ω′₀=0; p,p⁺=0, 1, . . . ,w_(2,1)/w_(2,2), p⁺≠0 if p=1 and u=1, w_(2,1), w_(2,2)=1,2, . . . ,w_(2,1)≠w_(2,2), q=2, 3 . . . , q_(R)=0 for q even, 1 for q odd whereφ=Lkn, φ⁺=Lk⁺n, k=2π/λ, k⁺=2π[(1/λ)+(ω/2πc)], wherein ω is the angularfrequency splitting between the two beams, ω′_(u′) are frequencies notequal to ω caused by at least one of the detector, the analyzer, and asource for the two beams, L is the physical path length difference, λ isthe wavelength of the beams in the first set, n is a refractive index, cis the speed of light in vacuum, and t is time, and wherein the dominantterm corresponds to a_(1,0,1,0) cos(ωt+φ+ζ_(1,0,1,0)) and the additionalterms correspond to the remaining terms, and wherein the amplitudesa_(v) and B_(v) and phases ζ_(v) define the coefficients for therepresentation of s(t), the subscript v denoting a general index.
 9. Thesystem of claim 8, wherein to quantify the at least one additional term,the analyzer calculates a frequency spectrum corresponding to a set ofthe values of s(t) and estimates the coefficients for the at least oneadditional term based on the amplitude and phase of the frequencyspectrum at an angular frequency {tilde over (ω)} equal to thederivative with respect to time of the argument of one of the sinusoidsin the representation of s(t) not corresponding to the dominant term, oran alias {tilde over (ω)}_(A) of {tilde over (ω)}.
 10. The system ofclaim 9, wherein the frequency spectrum is the Fourier transform of theset of values of s(t).
 11. The system of claim 9, wherein s(t) can beexpressed as s(t)=A(t)cos(α(t)), α(t) being the phase of s(t), and thefrequency spectrum is the Fourier transform of α(t).
 12. The system ofclaim 9, wherein the detector has a sampling rate that defines a Nyquistfrequency ω_(Ny), and wherein the analyzer estimates the coefficientsfor the at least one additional term based on the amplitude and phase ofthe frequency spectrum at the alias {tilde over (ω)}_(A) of {tilde over(ω)}, wherein {tilde over (ω)}_(A)=(−1)^(r){tilde over(ω)}−[(−1)^(r)(r+(½))−(½)]ω_(Ny) for a positive integer of r thatsatisfies rω _(Ny)<{tilde over (ω)}<(r+1)ω_(Ny).
 13. The system of claim9, wherein {tilde over (ω)} is one of ω+ω′_(u′) for u′≠0.
 14. The systemof claim 9, wherein {tilde over (ω)} is one of q(ω+{dot over (φ)}). 15.The system of claim 9, wherein {tilde over (ω)} is one of uω+p{dot over(φ)}+p⁺{dot over (φ)}, for p≠1 and, p≠0 when u=0.
 16. The system ofclaim 9, wherein to estimate the coefficients for the at least oneadditional term the analyzer normalizes the amplitude and phase of thefrequency spectrum at the angular frequency {tilde over (ω)} to accountfor at least one non-zero, derivative of {dot over (φ)}.
 17. The systemof claim 1, wherein the analyzer quantifies the at least one additionalterm based on a first set of values of s(t) for which the Doppler shiftis sufficiently large to spectrally separate the additional frequencyfrom the dominant frequency, and further quantifies the at least oneadditional term based on at least a second set of values of s(t) forwhich the Doppler shift is different from that of the first set ofvalues and sufficiently large to spectrally separate the additionalfrequency from the dominant frequency.
 18. The system of claim 17,wherein the analyzer quantifies the at least one additional term as afunction of the Doppler shift by interpolating values of thequantification for each set of values of s(t).
 19. The system of claim9, wherein the analyzer determines the dependence of each of theestimated coefficients on the Doppler shift based on multiple sets ofvalues of s(t), each set corresponding to a different Doppler shift. 20.The system of claim 9, wherein the at least one additional term is aplurality of the additional terms and wherein to quantify the pluralityof the additional terms the analyzer estimates the coefficients for eachof the plurality of the additional terms based on the amplitude andphase of the frequency spectrum at a corresponding plurality of angularfrequencies {tilde over (ω)}_(v) or their aliases, wherein each {tildeover (ω)}_(v) equals the derivative with respect to time of the argumentof one of the sinusoids in the representation of s(t) not correspondingto the dominant term.
 21. The system of claim 20, wherein the analyzerestimates coefficients corresponding to at least some ofB_(1,0,1,0,q,q−2j) and ζ_(1,0,1,0,q,q−2j), where q is odd and j is anonnegative integer less than q/2−1, to determine B_(1,0,1,0,q,1) andζ_(1,0,1,0,q,1).
 22. The system of claim 1, wherein the analyzerestimates the change in the optical path length difference correspondingto the other value of s(t) by determining a value for φ=Lkn that isself-consistent with s(t)∝cos(ωt+φ+ζ_(1,0,1,0))+NL(φ,{dot over (φ)}),where NL expresses the quantified at least one additional term, whereinL is the physical path length difference, k is a wavenumber, n is arefractive index, ω is the angular frequency difference between the twobeams, t is time, and ζ_(1,0,1,0) is a phase-offset.
 23. The system ofclaim 22, wherein the analyzer determines the value for φ by iterativelyimproving an estimate for the value for φ.
 24. The system of claim 1,wherein during operation the analyzer uses the estimated change inoptical path length to determine a change in physical path length. 25.The system claim 1, wherein during operation the analyzer uses theestimated change in optical path length to determine a change indispersion.
 26. The system of claim 1, wherein during operation theanalyzer uses the estimated change in optical path length to determinean intrinsic value a gas.
 27. The system of claim 1, wherein duringoperation the analyzer uses the estimated change in optical path lengthto monitor the wavelength of the beams.
 28. An interferometry systemcomprising: an interferometer which during operation directs two beamsalong separate paths and then combines the beams to produce anoverlapping pair of exit beams, the separate paths defining an opticalpath length difference; a detector which responds to opticalinterference between the overlapping pair of exit beams and produces asignal s(t) indicative of the interference, the signal s(t) being afunction of the optical path length difference, wherein properties ofthe interferometry system cause the signal s(t) to deviate from theexpression s(t)=a cos(ωt+φ+ζ) , where φ=Lkn, L is the physical pathlength difference, k is a wavenumber, n is a refractive index, ω is theangular frequency difference, if any, between the two beams, t is time,a is an amplitude that is constant with respect to φ, and ζ is a phaseoffset that is constant with respect to φ and {dot over (φ)}; ananalyzer coupled to the detector which during operation: i) Fouriertransforms at least one set of values of s(t) for which the rate ofchange of the optical path length difference is not zero ({dot over(φ)}≠0), the Fourier transform defining a power spectrum equal to thesquare modulus of the Fourier transform; ii) quantifies at least some ofthe deviations based on the amplitude and phase of the Fourier transformat frequencies that differ from ω+{dot over (φ)} and correspond to peaksin the power spectrum; and iii) uses the quantified deviations toestimate a change in the optical path length difference corresponding toa particular value of s(t).
 29. An interferometry system comprising: aninterferometer which during operation directs two beams along separatepaths and then combines the beams to produce an overlapping pair of exitbeams, the separate paths defining an optical path length difference; adetector which responds to optical interference between the overlappingpair of exit beams and produces a signal s(t) indicative of theinterference, the signal s(t) being a function of the optical pathlength difference, wherein properties of the interferometry system causethe signal s(t) to deviate from the expression s(t)=a cos(ωt+φ+ζ), whereφ=Lkn, L is the physical path length difference, k is a wavenumber, n isa refractive index, ω is the angular frequency difference, if any,between the two beams, t is time, a is an amplitude that is constantwith respect to φ, and ζ is a phase offset that is constant with respectto φ and {dot over (φ)}, and wherein s(t) can be expressed ass(t)=A(t)cos(α(t)), α(t) being the phase of s(t); an analyzer coupled tothe detector which during operation: i) extracts the phase α(t) fors(t); ii) Fourier transforms at least one set of values of α(t) forwhich the rate of change of the optical path length difference is notzero ({dot over (φ)}≠0), the Fourier transform defining a power spectrumequal to the square modulus of the Fourier transform; iii) quantifies atleast some of the deviations based on the amplitude and phase of theFourier transform at frequencies that differ from ω+{dot over (φ)} andcorrespond to peaks in the power spectrum; and iv) uses the quantifieddeviations to estimate a change in the optical path length differencecorresponding to a particular value of s(t).
 30. An interferometrysystem comprising: an interferometer which during operation directs twobeams along separate paths and then combines the beams to produce anoverlapping pair of exit beams, the separate paths defining an opticalpath length difference; a detector which responds to opticalinterference between the overlapping pair of exit beams and produces aninterference signal s(t) indicative of the optical path lengthdifference, the signal s(t) including a dominant term having a frequencyequal to the sum of the frequency splitting ω between the two beams, ifany, and a Doppler shift {dot over (φ)} defined by the rate of change ofthe optical path length difference, wherein properties of theinterferometry system causes the signal s(t) to further includeadditional terms each having a frequency not equal to the sum of thefrequency splitting ω and the Doppler shift {dot over (φ)}; an analyzercoupled to the detector which during operation monitors the frequenciesof the signal s(t), and produces a signal indicative of systemdegradation when the amplitude of a frequency corresponding to one ofthe additional terms exceeds a threshold value; and an alert mechanismcoupled to the analyzer and responsive to the system degradation signal.31. The system of claim 30, wherein the alert mechanism comprises atleast one of a visual display, a sound system, a warning light, and aprinter.
 32. The system of claim 30, wherein during operation theanalyzer monitors the frequencies in s(t) by Fourier transforming atleast one set of values for s(t).
 33. The system of claim 30, whereins(t) can be expressed as s(t)=A(t)cos(α(t)), α(t) being the phase ofs(t), and wherein during operation the analyzer monitors the frequenciesin s(t) by extracting the phase α(t) from at least one set of values fors(t) and Fourier transforming the extracted phases α(t).
 34. The systemof claim 30, wherein during operation the analyzer monitors thefrequencies in s(t) based on values of s(t) for which the value of theDoppler shift causes the dominant term and at least one of theadditional terms to be separated spectrally.
 35. The system of claim 30,wherein s(t) can be expressed as${s(t)} = {{a_{1,0,1,0}{\cos \left( {{\omega \quad t} + \phi + \zeta_{1,0,1,0}} \right)}} + {\sum\limits_{u,u^{\prime},p,p^{+}}{a_{u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega \quad t} + {\omega_{u^{\prime}}^{\prime}t} + {p\quad \phi} - {p^{+}\phi^{+}} + \zeta_{u,u^{\prime},p,p^{+}}} \right)}}} + {\sum\limits_{q}{\left( a_{1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{1,0,1,0,q,q}{\cos \left\lbrack {{q\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{1,0,1,0,q,{q - 2}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}}}$

+ . . . , u=0 or 1; u′=0,1, . . . ; ω′₀=0; p,p⁺=0, 1, . . . ,w_(2,1)/w_(2,2), p⁺≠0 if p=1 and u=1, w_(2,1), w_(2,2)=1,2, . . . ,w_(2,1)≠w_(2,2), q=2, 3 . . . , q_(R)=0 for q even, 1 for q odd whereφ=Lkn, φ⁺=Lk⁺n, k=2π/λ, k⁺=2π[(1/λ)+(ω/2πc)], and wherein ω is theangular frequency splitting between the two beams, ω′_(u′) arefrequencies not equal to ω caused by at least one of the detector, theanalyzer, and a source for the two beams, L is the physical path lengthdifference, λ is the wavelength of the beams in the first set, n is arefractive index, c is the speed of light in vacuum, and t is time, andwherein the additional terms correspond to the terms other thana_(1,0,1,0) cos(ωt+φ+ζ_(1,0,1,0)).
 36. The system of claim 35, whereinduring operation the analyzer compares the amplitude of one offrequencies ω+ω′_(u′), for u′≠0, to the threshold value to determinewhether to produce the signal indicative of system degradation.
 37. Thesystem of claim 35, wherein during operation the analyzer compares theamplitude of one of frequencies q(ω+{dot over (φ)}) to the thresholdvalue to determine whether to produce the signal indicative of systemdegradation.
 38. The system of claim 35, wherein during operation theanalyzer compares the amplitude of frequencies at one of uω+p{dot over(φ)}+p⁺{dot over (φ)}, for p≠1, and for p≠0 when u=0, to the thresholdvalue to determine whether to produce the signal indicative of systemdegradation.
 39. An interferometry system comprising: a source whichduring operation provides a first set of two beams having a frequencysplitting ω and a second set of two beams having a frequency splittingω_(T) not equal to ω; an interferometer which during operation directsthe first beam of the first set and the first beam of the second setalong a measurement path and the second beam of the first set and thesecond beam of the second set along a reference path, and then combinesthe two sets of beams to form an output beam, the measurement andreference paths defining an optical path length difference; a detectorwhich responds to optical interference between the beams in the outputbeam and produces a signal S(t) indicative of the interference, theinterference being a function of the optical path length difference; andin the absence of the second set of beams, the signal S(t) equals s(t)which includes a dominant term at a frequency equal to the sum of thefrequency splitting ω and a Doppler shift {dot over (φ)} defined by therate of change of the optical path length difference, wherein propertiesin the interferometry system cause zero-frequency-shift cyclic errorsthat contribute to s(t) at the same frequency as the dominant frequency,in the presence of the second set of beams, the properties that producethe zero-frequency-shift cyclic error contribution to s(t) produce amultiplet in the frequency spectrum of S(t), wherein the multiplet hasadjacent peaks that are spaced by ω−ω_(T); an analyzer coupled to thedetector which during operation resolves frequencies in S(t) to identifythe multiplet and quantifies at least one of the zero-frequency-shiftcyclic errors based on the amplitude and phase of at least one of thepeaks in the multiplet.
 40. The system of claim 39, wherein duringoperation the analyzer quantifies multiple zero-frequency-shift cyclicerrors based on the amplitude and phase of each of multiple peaks in themultiplet.
 41. The system of claim 39, wherein the analyzer is furthercoupled to the source and during operation selectively causes the sourceto provide the first set of beams and not the second set of beams to theinterferometer.
 42. The system of claim 41, wherein during operationwhen the analyzer selectively causes the source to provide the first setof beams and not the second set of beams to the interferometer, theanalyzer determines the optical path length difference based on s(t) andat least one of the quantified zero-frequency-shift cyclic errors. 43.The system of claim 39, wherein during operation the analyzer resolvesfrequencies in S(t) by Fourier transforming at least one set of valuesfor S(t).
 44. The system of claim 39, wherein S(t) can be expressed asS(t)=A_(S)(t)cos(α_(S)(t)), α_(S)(t) being the phase of S(t), andwherein during operation the analyzer resolves the frequencies of S(t)byextracting the phase α_(S)(t) from S(t) and Fourier transforming atleast one set of values of α_(S)(t).
 45. The system of claim 39, whereinthe multiplet includes a peak at the dominant frequency.
 46. The systemof claim 39, wherein the detector samples values of S(t) at a rate thatdefines a Nyquist frequency, and the frequency splittings are each lessthan the Nyquist frequency.
 47. The system of claim 39, wherein thedetector samples values of S(t) at a rate that defines a Nyquistfrequency and the difference between the average frequency of the firstset of beams and the average frequency of the second set of beams ismore than the Nyquist frequency.
 48. The system of claim 39, wherein thedetector samples values of S(t) at a rate that defines a Nyquistfrequency ω_(Ny), and ω<ω_(Ny), ω_(T)<ω_(Ny), and |ω−ω_(T)|<<ω.
 49. Thesystem of claim 48, wherein |ω−ω_(T)|<(ω/100).
 50. The system of claim39, wherein the source comprises first and second lasers, the first setof beams derived from the first laser and the second set of beamsderived from the second laser.
 51. The system of claim 39, wherein thesource comprises first and second lasers and first and secondacousto-optical modulators, the first set of beams derived from thefirst laser and the first acousto-optical modulator and the second setof beams derived from second laser and the second acousto-opticalmodulator.
 52. The system of claim 39, wherein the source comprises alaser and first and second acousto-optical modulators, a first beamderived from the laser passes through first acousto-optical modulator toproduce the first set of beams and a second beam derived from the laserpasses through the second acousto-optical modulator to produce thesecond set of beams.
 53. The system of claim 52, wherein the first andsecond beams derived from the laser correspond to adjacent longitudinalmodes of the laser.
 54. The system of claim 39, wherein during operationthe analyzer resolves the frequency multiplet in S(t) for each ofmultiple Doppler shifts and quantifies the dependence of the quantifiedzero-frequency-shift cyclic on the Doppler shift.
 55. The system ofclaim 39, wherein during operation the analyzer produces a signalindicative of system degradation when the amplitude of the multipletexceeds a threshold value, and the system further includes an alertmechanism coupled to the analyzer and responsive to the systemdegradation signal.
 56. The system of claim 55, wherein the alertmechanism comprises at least one of a visual display, a sound speaker, aprinter, and a warning light.
 57. The system of claim 39, wherein s(t)can be expressed as${{s(t)} = {{a_{1,0,1,0}{\cos \left( {{\omega \quad t} + \phi + \zeta_{1,0,1,0}} \right)}} + {\sum\limits_{u,u^{\prime},p,p^{+}}{a_{u,u^{\prime},p,p^{+}}{\cos \left( {{u\quad \omega \quad t} + {\omega_{u^{\prime}}^{\prime}t} + {p\quad \phi} - {p^{+}\phi^{+}} + \zeta_{u,u^{\prime},p,p^{+}}} \right)}}} + {\sum\limits_{q}{\left( a_{1,0,1,0} \right)^{q}\begin{Bmatrix}{{B_{1,0,1,0,q,q}{\cos \left\lbrack {{q\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q}} \right\rbrack}} +} \\{{B_{1,0,1,0,q,{q - 2}}{\cos \left\lbrack {{\left( {q - 2} \right)\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,{q - 2}}} \right\rbrack}} +} \\{\ldots +} \\{B_{1,0,1,0,q,q_{R}}{\cos \left\lbrack {{q_{R}\left( {{\omega \quad t} + \phi} \right)} + \zeta_{1,0,1,0,q,q_{R}}} \right\rbrack}}\end{Bmatrix}}} + \ldots}},$

u=0 or 1; u′=0,1, . . . ; ω′₀=0; p,p⁺=0, 1, . . . , w_(2,1)/w_(2,2),p⁺≠0 if p=1 and u=1, w_(2,1), w_(2,2)=1,2, . . . , w_(2,1)≠w_(2,2), q=2,3 . . . , q_(R)=0 for q even, 1 for q odd where φ=Lkn, φ⁺=Lk⁺n, k=2π/λ,k⁺=2π[(1/λ)+(ω/2πc)], and where ω′_(u′) are frequencies not equal to ωcaused by at least one of the detector, the analyzer, and the source, Lis the physical path length difference, λ is the wavelength of the beamsin the first set, n is a refractive index, c is the speed of light invacuum, and t is time, wherein the nonlinearities cause terms other thana_(1,0,1,0) cos(ωt+φ+ζ_(1,0,1,0)), and wherein the quantifiedzero-frequency-shift cyclic error corresponds to B_(1,0,1,0,q,1) andζ_(1,0,1,0,q,1) for one of q=3,5,7 . . . .
 58. A lithography system foruse in fabricating integrated circuits on a wafer, the systemcomprising: a stage for supporting the wafer; an illumination system forimaging spatially patterned radiation onto the wafer; a positioningsystem for adjusting the position of the stage relative to the imagedradiation; and the interferometry system of claim 1, 28, 29, 30, or 39for measuring the position of the stage.
 59. A lithography system foruse in fabricating integrated circuits on a wafer, the systemcomprising: a stage for supporting the wafer; and an illumination systemincluding a radiation source, a mask, a positioning system, a lensassembly, and the interferometry system of claim 1, 28, 29, 30, or 39wherein during operation the source directs radiation through the maskto produce spatially patterned radiation, the positioning system adjuststhe position of the mask relative to the radiation from the source, thelens assembly images the spatially patterned radiation onto the wafer,and the interferometry system measures the position of the mask relativeto the radiation from the source.
 60. A beam writing system for use infabricating a lithography mask, the system comprising: a sourceproviding a write beam to pattern a substrate; a stage supporting thesubstrate; a beam directing assembly for delivering the write beam tothe substrate; a positioning system for positioning the stage and beamdirecting assembly relative one another; and the interferometry systemof claim 1, 28, 29, 30, or 39 for measuring the position of the stagerelative to the beam directing assembly.
 61. An interferometry methodfor use with an interferometry system comprising: directing two beamsalong separate paths; combining the beams to produce an overlapping pairof exit beams, the separate paths defining an optical path lengthdifference; measuring optical interference between the overlapping pairof exit beams to produce an interference signal s(t) indicative of theoptical path length difference, the signal s(t) including a dominantterm having a frequency equal to the sum of the frequency splitting ωbetween the two beams, if any, and a Doppler shift {dot over (φ)}defined by the rate of change of the optical path length difference,wherein properties of the interferometry system causes the signal s(t)to further include additional terms each having a frequency not equal tothe sum of the frequency splitting ω and the Doppler shift {dot over(φ)}; quantifying at least one of the additional terms based on valuesof s(t) for which the value of the Doppler shift causes the dominantterm and the at least one additional term to be separated spectrally;and using the quantified at least one additional term to estimate achange in the optical path length difference corresponding to anothervalue of s(t) for which the value of the Doppler shift does not causesthe dominant term and the at least one additional term to overlapspectrally.
 62. An interferometry method for use with an interferometrysystem comprising: directing two beams along separate paths; combiningthe beams to produce an overlapping pair of exit beams, the separatepaths defining an optical path length difference; measuring opticalinterference between the overlapping pair of exit beams to produce aninterference signal s(t) indicative of the optical path lengthdifference, the signal s(t) including a dominant term having a frequencyequal to the sum of the frequency splitting ω between the two beams, ifany, and a Doppler shift {dot over (φ)} defined by the rate of change ofthe optical path length difference, wherein properties of theinterferometry system causes the signal s(t) to further includeadditional terms each having a frequency not equal to the sum of thefrequency splitting ω and the Doppler shift {dot over (φ)}; monitoringthe frequencies of the signal s(t); and alerting an operator when theamplitude of a frequency corresponding to one of the additional termsexceeds a threshold value.
 63. An interferometry method for use with aninterferometry system comprising: providing a first set of two beamshaving a frequency splitting ω and a second set of two beams having afrequency splitting ω_(T) not equal to ω; directing the first beam ofthe first set and the first beam of the second set along a measurementpath and the second beam of the first set and the second beam of thesecond set along a reference path; combining the two sets of beams toform an output beam, the measurement and reference paths defining anoptical path length difference; measuring optical interference betweenthe beams in the output beam to produce a signal S(t) indicative of theinterference, the interference being a function of the optical pathlength difference, wherein in the absence of the second set of beams,the signal S(t) equals s(t) which includes a dominant term at afrequency equal to the sum of the frequency splitting ω and a Dopplershift {dot over (φ)} defined by the rate of change of the optical pathlength difference, wherein properties in the interferometry system causezero-frequency-shift cyclic errors that contribute to s(t) at the samefrequency as the dominant frequency, in the presence of the second setof beams, the properties that produce the zero-frequency-shift cyclicerror contribution to s(t) produce a multiplet in the frequency spectrumof S(t), wherein the multiplet has adjacent peaks that are spaced byω−ω_(T); resolving frequencies in S(t) to identify the multiplet; andquantifying at least one of the zero-frequency-shift cyclic errors basedon the amplitude and phase of at least one of the peaks in themultiplet.
 64. A lithography method comprising: supporting a wafer on astage; imaging spatially patterned radiation onto the wafer; adjustingthe position of the stage relative to the imaged radiation; and usingthe interferometry method of claim 61, 62, or 63 to measure the relativeposition of the stage.
 65. A lithography method comprising: supporting awafer on a stage; directing radiation from a source through a mask toproduce spatially patterned radiation; positioning the mask relative tothe radiation; using the interferometry method of claim 61, 62, or 63 tomeasures the position of the mask relative to the radiation; and imagingthe spatially patterned radiation onto the wafer.
 66. A beam writingmethod comprising: providing a write beam to pattern a substrate;supporting the substrate on a stage; delivering the write beam to thesubstrate; positioning the stage relative to the write beam; and usingthe interferometry method of claim 61, 62, or 63 to measure the relativeposition of the stage.